# Models of Function Type for Commutative Symmetric Operator Families in Krein Spaces

Authors: Vladimir Strauss
Journal: Abstract and Applied Analysis
 (2008)

---

## Abstract

Commutative symmetric operator families of the so-calledD
_
+-class are considered in Krein spaces. It is proved that the restriction of a family of this type on a special kind of invariant subspace is similar to a family of operators adjoint to multiplication operators by scalar functions acting on a suitable function space.


---

## Body

## 1. Introduction

In the
well-known monograph of Colojoară and Foiaş [1], it is mentioned (in Chapter
5, Corollary 5.7) that J-unitary and J-self-adjoint
operators in Pontrjagin spaces give examples of generalized spectral operators.
Actually, it is clear that this observation can be applied also to many operator
classes in Krein spaces, so Operator Theory in Krein spaces is a good
laboratory where different methods of general Operator Theory can be tested.
One of these methods is related with model spaces of function type.The main goal of the paper is a problem of model
representation for a commutative operator family𝔜 acting on a
separable Krein space and possessing a maximal nonnegative invariant subspace,
presented as a direct sum of a neutral subspace with a finite dimension and a
uniformly positive subspace. As it is known (see [2]), this family generates a
spectral function E
λ with a peculiar
spectral set Λ that provides a
resolution of the spectral type for the family. In particular, with every
operator A, one can associate a scalar function f
A
(
λ
) such that(1.1)
A
E
(
Δ
)
=
∫
Δ
f
A
(
λ
)
d
E
λ
,where Δ runs through
the set of all closed intervals of the real line disjoint with Λ.The key results
(Theorems6.5, 6.19, and 6.29) say that there is a suitable function space
(so-called basic model space) where f
A
(
λ
) generates a
multiplication operator similar to a compression of A on a subspace
calculated through E
λ.Section2 gives
a draft of problems considered in the paper, analyzing a simple case of
self-adjoint operator in Pontryagin space. Section 3 contains definitions and
well-known results used throughout the paper. Section 4 deals with a model
representation of a resolution of the identity that is simultaneously J-orthogonal and
similar to an orthogonal resolution of the identity. In Section 5 a notion of
unbounded elements conformed with a resolution of the identity is introduced
and studied. Roughly speaking, a resolution sets a correspondence between a
Hilbert space and a (vector-valued) function space L
σ
→2. From this point of view, the unbounded elements
correspond to measurable vector-valued functions outside of L
σ
→2. The main results are presented in Section 6, where
the notion of a basic model space is introduced. Also, a relation is
established between multiplication operators by scalar functions acting in a
basic model space, and operators of a commutative J-symmetric
algebra of D
κ
+-class. Here,
emphasis is made on the problem of uniqueness of a basic model space for such
algebras. Section 6.2 corresponds to resolutions of the identity with
properties like those in Pontryagin spaces. Section 6.3 deals with general J-orthogonal
resolutions of the identity of D
κ
+-class, and
Section 6.4 contains a theorem on a model representation of a commutative J-symmetric
operator family of D
κ
+-class.
Historical and bibliographical remarks are presented in the last section.It is assumed
that the reader is familiar with elements of Krein space geometry and Operator
theory (see [3–7]). In this paper, the terminology given in [8] will be used.


## 2. An Elementary Description of the Problem

Letℋ be a separable
Hilbert space with a scalar product (
⋅
,
⋅
). ℋ is said to be an indefinite metric space if it is
equipped by a sesquilinear continuous Hermitian form (indefinite inner product) [
⋅
,
⋅
] such that the
corresponding quadratic form has an indefinite sign (i.e., [
x
,
x
] takes positive,
negative, and zero values). The indefinite inner product can be represented in
the form [
⋅
,
⋅
]
=
(
G
⋅
,
⋅
), where G is a so-called
Gram operator. The operator G is bounded and
self-adjoint. If the Gram operator for an indefinite metric space is boundedly
invertible and its invariant subspace that corresponds to the negative spectrum
(or, alternatively, to the positive spectrum) of G is finite
dimensional, the space is called a Pontryagin space. In this section, we
consider Pontryagin spaces only.Now, letA: ℋ
↦
ℋ be a bounded
linear operator. This operator is said to be π-self-adjoint (π-s.a.) if [
A
x
,
y
]
=
[
x
,
A
y
] for all x
,
y
∈
ℋ. The properties of π-s.a. operators
differ from the properties of ordinary self-adjoint operators. For instance, a π-s.a. can have
a nonreal spectrum or/and a nontrivial Jordan chain. If A has only a real
spectrum, then there is, on ℝ, the eigen spectral function (e.s.f.) E
λ of A with a finite
set Λ of critical
points such that (Δ
=
(
a
;
b
)
,
a
,
b
∉
Λ)
(2.1)
(
a
)
for every
λ
∈
ℝ
∖
Λ
⁢
it holds that
A
E
λ
=
E
λ
A
;
(
b
)
σ
(
A
∣
E
(
Δ
)
ℋ
)
⊂
Δ
¯
;
(
c
)
if
Δ
∩
Λ
=
∅
then the subspace
E
(
Δ
)
ℋ
⁢
is positive
;
(
d
)
if
Δ
∩
Λ
≠
∅
then the subspace
E
(
Δ
)
ℋ
⁢
is negative or indefinite
.Let λ
0
∈
Λ. Then for every point of Λ there are three
options.(1)
For an intervalΔ
=
(
a
;
b
) such that Δ
∩
Λ
=
{
λ
0
} and a
,
b
∉
Λ, the spectral function E
λ
|
E
(
Δ
)
ℋ is bounded and
the representation ∫
Δ
ξ
E
(
d
ξ
) takes place.
(2)
For an intervalΔ
=
(
a
;
b
) such that Δ
∩
Λ
=
{
λ
0
} and a
,
b
∉
Λ, the spectral function E
λ
|
E
(
Δ
)
ℋ is bounded but
the representation ∫
Δ
ξ
E
(
d
ξ
) does not take
place.
(3)
For an intervalΔ
=
(
a
;
b
) such that Δ
∩
Λ
=
{
λ
0
} and a
,
b
∉
Λ, the spectral function E
λ
|
E
(
Δ
)
ℋ is unbounded.The first item
means that the operatorA
|
E
(
Δ
)
ℋ is similar to a
self-adjoint operator, that is, it is a scalar spectral operator. The second
item corresponds to a spectral operator with a nontrivial finite-dimensional
nilpotent summand. Thus only the third item represents a situation that is out
of the ordinary theory of model representation for spectral operators with real
spectrum. Below, we consider this exceptional case. Let us give an example of
the case in discussion.Example 2.1.
Let the systemh
,
g
,
e
1
,
e
2
,
e
3
,
… be an
orthonormalized basis in a Hilbert space ℋ and let [
x
,
y
]
=
(
x
,
h
)
(
g
,
y
)
+
(
x
,
g
)
(
h
,
y
)
+
∑
j
=
1
∞
(
x
,
e
j
)
(
e
j
,
y
). Then this space is a Pontryagin space with κ
=
1. Let an operator A be given by the
conditions (i)
A
h
=
0,
(ii)
A
e
j
=
1
/
j
⋅
(
e
j
+
h
), j
=
1
,
2
,
…,
(iii)
A
g
=
h
+
∑
j
=
1
∞
(
1
/
j
)
⋅
e
j. These
conditions define a bounded operator, so A is naturally
defined on ℋ and, moreover,
is a π-self-adjoint operator.
The spectral function of A can be
described by the following conditions: (2.2)
E
(
{
1
j
}
)
x
=
[
x
,
(
e
j
+
h
)
]
⋅
(
e
j
+
h
)
, so (2.3)
E
(
{
1
j
}
j
=
1
n
)
x
=
∑
j
=
1
n
[
x
,
(
e
j
+
h
)
]
⋅
(
e
j
+
h
)
. Put x
n
=
1
/
n
⋅
∑
j
=
1
n
e
j. Then E
(
{
1
/
j
}
j
=
1
n
)
x
n
=
1
/
n
⋅
∑
j
=
1
n
(
e
j
+
h
)
=
h
+
x
n. At the same time ∥
x
n
∥
=
1
/
n
→
0 for n
→
∞. Thus, for A and its
spectral function E
λ, we have the following results:(i)
E
λ is unbounded;
(ii)
h belongs to the
closure of the linear span ℋ
˜ generated by
the subspaces E
(
Δ
)
ℋ, where Δ
=
(
a
;
b
), 0
<
a
<
b, so the kernel of the operator A
|
ℋ
˜ is not trivial;
(iii)
ℋ
˜ is
degenerated, ℋ
˜
∩
ℋ
˜
[
⊥
]
=
{
μ
h
}
μ
∈
ℂ.Let us note
that the coexistence of these properties reflects a general situation (in particular, the first item yields the other two). Indeed,
it is shown by the following theorem.Theorem 2.2 (see [8, Theorem I.9.6]).
Let a subspaceℒ be positive.
Then there is a constant c
>
1 such that (
x
,
x
)
≤
c
[
x
,
x
] for every x
∈
ℒ.For simplicity
everywhere below in this section, we assume that(2.4)
(a)
Λ
=
{
0
}
,
(b)
E
λ
⁢
is unbounded
,
(c)
for every interval
Δ
=
[
a
;
b
]
,
such that
0
∉
Δ
⁢
the operator
A
|
E
(
Δ
)
ℋ
has simple spectrum
.Then by Theorem 2.2, the subspace ℋ
˜ generated as
the closed linear span of the subspaces E
(
Δ
)
ℋ, where Δ
=
[
a
;
b
], 0
∉
Δ, is not a positive subspace, thus it has the
nontrivial isotropic part, ℋ
˜
∩
ℋ
˜
[
⊥
]
≠
{
0
}. Let us denote, below, that ℋ
1
:
=
ℋ
˜
∩
ℋ
˜
[
⊥
]. Since the subspace ℋ
˜ is evidently
nonnegative, it can be presented in the form(2.5)
ℋ
˜
=
ℋ
1
∔
ℋ
2
,where ℋ
2 is a positive
subspace. Note that ℋ
2 is not uniquely
defined. Note also that, due to the general theory, the subspace ℋ
˜ is a part of a
maximal nonnegative subspace ℒ
+ invariant with
respect to A.Theorem 2.3.
LetE
λ be the e.s.f.
of a π-self-adjoint
operator A. Then there are a scalar Lebesgue measure μ
σ, the Hilbert space L
σ
2 associated with μ
σ, and a collection of μ
σ-measurable
scalar functions {
g
˜
j
(
t
)
}
j
=
1
k such that if Δ
=
[
a
,
b
]
,
0
∉
Δ, the operator E
(
Δ
)
|
ℋ
˜ is similar to
the operator acting by the formulae(2.6)
f
(
t
)
⟼
χ
Δ
(
t
)
f
(
t
)
+
∑
j
=
1
k
∫
Δ
f
(
t
)
g
˜
j
(
t
)
¯
d
σ
(
t
)
⋅
e
j
,
e
j
⟼
0
j
=
1
,
2
,
…
,
k
on the formal linear span of L
σ
2 and {
e
j
}
j
=
1
k, where {
e
j
}
j
=
1
k is a basis on ℋ
1, χ
Δ
(
t
) is the
indicator of Δ, χ
Δ
(
t
)
g
˜
j
(
t
)
∈
L
σ
2 for j
=
1
,
2
,
…
,
n, and(2.7)
∑
j
=
1
k
α
j
g
˜
j
(
t
)
∈
L
σ
2
⟹
α
1
=
α
2
=
⋯
=
α
k
=
0.
Proof.
According to Equality (2.5), the
restriction E
(
Δ
)
|
ℋ
˜ has the
following matrix representation:(2.8)
E
(
Δ
)
|
ℋ
˜
=
(
0
E
12
(
Δ
)
0
E
22
(
Δ
)
)
.With no loss of generality (see
Proposition 3.3 below for details), one can assume that (
⋅
,
⋅
)
=
[
⋅
,
⋅
] on ℋ
2 and ℋ
1
⊥
ℋ
2. Then the operator-valued function F
λ
:
=
E
22
(
(
−
∞
,
λ
)
) represents an
(orthogonal) resolution of the identity acting in the Hilbert space ℋ
2. From the classical theory, we know that there is a
Hilbert space L
σ
2 of scalar
functions that is a model space for F
λ with some
corresponding operator W: L
σ
2
↦
ℋ
2 of similarity,
that is,(2.9)
W
−
1
E
22
(
Δ
)
W
f
(
t
)
=
χ
Δ
(
t
)
f
(
t
)
.Let {
e
j
}
1
n be an
orthonormalized basis in ℒ
1. Then, for every Δ
=
[
a
,
b
], 0
∉
Δ, and j
=
1
,
2
,
…
n, the expression ν
j
Δ
W
f
(
t
)
:
=
(
E
12
(
Δ
)
W
f
(
t
)
,
e
j
) defines a
continuous linear functional in L
σ
2. Thus, for every j
=
1
,
2
,
…
n, there exists the F-measurable
function g
˜
j
(
t
) such that, for
an arbitrary Δ
=
[
a
,
b
], 0
∉
Δ, and f
(
t
)
∈
L
σ
2, the representation(2.10)
ν
j
Δ
W
f
(
t
)
=
∫
Δ
f
(
t
)
g
˜
(
t
)
¯
d
σ
(
t
)holds. Since the spectral
function E is unbounded, ∫
−
1
1
|
g
˜
(
t
)
|
2
d
σ
(
t
)
=
∞ for at least
one j
=
1
,
2
,
…
,
n. It means that at least one function (maybe all) of
the collection(2.11)
{
g
˜
j
(
t
)
}
j
=
1
nis not in L
σ
2. Then we just need to prove (2.7).
Let∑
j
=
1
n
α
j
⋅
g
˜
j
(
t
)
∈
L
σ
2. Then the linear functional (f
∈
L
σ
2)(2.12)
ϕ
f
:
=
∫
ℝ
f
(
t
)
∑
j
=
1
n
α
j
⋅
g
˜
j
(
t
)
¯
d
σ
(
t
)is bounded.
Next, by
definition, the spaceℋ
˜ is the closure
of all vectors having the form(2.13)
W
f
(
t
)
+
∑
j
=
1
k
∫
ℝ
f
(
t
)
g
˜
j
(
t
)
¯
d
σ
(
t
)
⋅
e
j
,with f
∈
L
σ
2 vanishing near
zero; so, for every x
∈
ℋ
1, there is a sequence {
f
m
}
m
=
1
∞ such that(2.14)
lim
⁡
m
→
∞
∥
f
m
∥
L
σ
2
=
0
,
lim
⁡
m
→
∞
∑
j
=
1
k
∫
ℝ
f
m
(
t
)
g
˜
j
(
t
)
¯
d
σ
(
t
)
⋅
e
j
=
x
.The latter yields(2.15)
(
x
,
∑
j
=
1
n
α
j
⋅
e
j
)
=
lim
⁡
m
→
∞
∑
j
=
1
k
∫
ℝ
f
m
(
t
)
α
j
g
˜
j
(
t
)
¯
d
σ
(
t
)
=
lim
⁡
m
→
∞
ϕ
f
m
=
0.Thus ∑
j
=
1
n
α
j
⋅
e
j
=
0, therefore α
1
=
α
2
=
⋯
=
α
n
=
0.Theorem 2.3 gives a
possibility to construct a partial model representation for A in a sense
exposed below. With no loss of generality, one can assume that the Hilbert
scalar product (
⋅
,
⋅
) on ℋ is such that
the corresponding Gram operator re-denoted as J (i.e., [
⋅
,
⋅
]
=
(
J
⋅
,
⋅
)) has the
properties J
=
J
−
1
=
J
∗. Let (see (2.5))(2.16)
ℋ
0
=
J
ℋ
1
.Then ℋ
0
⊥
ℋ
˜. Additionally, one can assume that the basis {
e
j
}
j
=
1
k from Theorem
2.3 is orthonormalized, [
⋅
,
⋅
]
=
(
⋅
,
⋅
) on ℋ
2 and ℋ
1
⊥
ℋ
2. Let P
0 and P
2 be
orthoprojections onto subspaces ℋ
0 and ℋ
2, respectively.Theorem 2.4.
Let a spaceL
σ
2 and functions {
g
˜
j
}
j
=
1
k be related with
the e.s.f E
λ of a π-s.a. operator A as in Theorem
2.3, and let L
˜
σ
2 be a Hilbert
space formed as the linear span of L
σ
2 and {
g
˜
j
}
j
=
1
k, where the functions in {
g
˜
j
}
j
=
1
k are mutually
orthogonal, have unit norm, and are orthogonal to the space L
σ
2 which conserves
its Hilbert structure. Then t
g
˜
j belongs to the
linear span of L
˜
σ
2 for every j
=
1
,
2
,
…
,
k, and the compression (
P
0
+
P
2
)
A
|
ℋ
0
⊕
ℋ
2 of A is similar to
the multiplication operator by t acting on L
˜
σ
2.Theorem2.4 is not new (see, e.g., a more
general result in [9]
and its applications in [10, 11])
but it opens some ways to generalization and generates some open questions and
problems. For instance, (2.4)(c) is restrictive and must be dropped. This, in
turn, implies a replacement of the space L
˜
σ
2 of scalar
functions by a suitable space of vector-valued functions. Next, the passing
from a single self-adjoint operator to a family of self-adjoint operators in
the case of indefinite metric spaces has some difficulties which must be taken
into account. Let us consider the following example.Example 2.5.
Let the
union of systems{
h
1
,
h
2
,
h
3
}, {
g
1
,
g
2
,
g
3
}, and {
e
k
}
k
=
1
∞ give an
orthonormalized basis in a Hilbert space ℋ. We define a fundamental symmetry J on ℋ by formulae(2.17)
J
g
j
=
h
j
,
J
h
j
=
g
j
,
j
=
1
,
2
,
3
,
J
e
k
=
e
k
,
k
=
1
,
2
,
…
.Thus ℋ becomes a
Pontryagin space. Next, we define operators A and B by formulae(2.18)
A
h
1
=
h
2
,
A
h
2
=
∑
k
=
1
∞
1
k
⋅
e
k
,
A
h
3
=
∑
k
=
1
∞
1
k
4
/
5
⋅
e
k
,
A
g
1
=
A
g
3
=
0
,
A
g
2
=
g
1
,
A
e
k
=
1
k
⋅
e
k
+
1
k
⋅
g
2
+
1
k
4
/
5
⋅
g
3
,
k
=
1
,
2
,
…
,
B
h
1
=
h
3
,
B
h
2
=
∑
k
=
1
∞
1
k
4
/
5
⋅
e
k
,
B
h
3
=
∑
k
=
1
∞
1
k
3
/
5
⋅
e
k
,
B
g
1
=
B
g
2
=
0
,
B
g
3
=
g
1
,
B
e
k
=
1
k
4
/
5
⋅
e
k
+
1
k
4
/
5
⋅
g
2
+
1
k
3
/
5
⋅
g
3
,
k
=
1
,
2
,
…
.
It is easy to check that the
operators A and B are π-self-adjoint
and commute. Next, the eigen spectral function (e.s.f.) of A has the form(2.19)
E
λ
=
{
0
,
if
λ
≤
0
;
I
−
∑
k
:
(
1
/
k
)
≥
λ
[
⋅
,
u
k
]
⋅
u
k
,
if
λ
∈
(
0
;
1
]
;
I
,
if
λ
>
1
,
where u
k
=
e
k
+
k
g
1
+
g
2
+
(
k
)
1
/
5
e
3, k
=
1
,
2
,
…. Moreover, for every β
>
α
>
0, we have the representations(2.20)
A
E
(
[
α
,
β
)
)
=
∫
α
β
λ
E
(
d
λ
)
,
B
E
(
[
α
,
β
)
)
=
∫
α
β
(
λ
)
4
/
5
E
(
d
λ
)
.Therefore, E
λ can be
considered as an “e.s.f.” of the family {
A
,
B
}; however,(2.21)
A
≠
∫
0
1
+
0
λ
E
(
d
λ
)
,
B
≠
∫
0
1
+
0
(
λ
)
4
/
5
E
(
d
λ
)
,because both integrals are
nonconvergent. Next, dim
Ker
(
A
)
=
dim
Ker
(
B
)
=
2, but Ker
(
A
)
≠
Ker
(
B
). Thus B
∉
Alg
(
A
) and A
∉
Alg
(
B
), where Alg
(
⋅
) means the
weakly closed algebra generated by the identity operator and the corresponding π-self-adjoint
operator. The same reasoning shows that there is no π-self-adjoint
operator C for which,
simultaneously, A
∈
Alg
(
C
) and B
∈
Alg
(
C
). Therefore, the description of the operator family in
discussion cannot be reduced to a representation of one operator. At the same
time the existence of the spectral function with (2.20) is not
occasional (see Theorem 3.11 below) and gives a possibility to construct a
partial model for the whole family.So one of the problems is to generalize Theorem2.4 to
commutative families of self-adjoint operators in Pontryagin and Krein spaces.
Next, it is clear that the space L
˜
σ
2 depends on the
choice of (2.5), but the latter is not uniquely determined. Thus,
another problem is a description of the ambiguity of L
˜
σ
2 or its analog
with vector-valued functions. The present paper is devoted mainly to these two
problems.Of course, the
direction marked by Theorem2.4 is not unique in relation to the goal to
construct a model representation of function type for operators in Pontryagin
spaces. In particular, in [12], it was shown that a bounded cyclic self-adjoint
operator A in a Pontryagin
space is unitary equivalent to the operator A
Φ of
multiplication by the independent variable in some space Π
(
Φ
) generated by a
“distribution” Φ. This distribution, in turn, is associated with a
quasiintegral representation for the moment sequence {
[
A
n
h
,
h
]
}
n
=
0
∞, where h is a cyclic
vector. The model in consideration describes the behavior of A on the whole
Pontryagin space, and in this sense, it is complete. On the contrary, the model
from Theorem 2.4 is a partial one; it describes only the part of A that can be
restored via its e.s.f.. It gives, indeed, an advantage to the model [12] but this advantage concerns
mainly cyclic operators because in the case of a non-cyclic operator the
corresponding space Π
(
Φ
) contains
matrix-valued functions, and this space has a more or less complicated
structure. The latter, however, is not the main obstacle. The experience with a
canonic representation of normal operators in finite-dimensional spaces with an
indefinite metric (see, e.g., [13, 14]) shows that any attempt to give an observable and,
simultaneously, complete description for a commutative family of self-adjoint
operators in spaces with indefinite metric with the rank of indefiniteness more
than 2 has little chances to prosper. On the contrary, a partial model
representation in the style of Theorem 2.4 can be generalized easily to
commutative families of π-self-adjoint
operators and to commutative families of J-self-adjoint
operators of the so-called D
κ
+-class. A
previous experience shows that, in spite of some incompleteness, this model was
useful in some applications concerning a single π-self-adjoint
operator, see Section 7 for more details.


## 3. Preliminaries

## 3.1. Krein Spaces

Letℋ be a Krein
space with an indefinite sesquilinear form [
⋅
,
⋅
], let ℋ
=
ℋ
+
[
+
˙
]
ℋ
− be its
canonical decomposition, let P
+ and P
− be canonical
projections ℋ
+
=
P
+
ℋ and ℋ
−
=
P
−
ℋ, let J
=
P
+
−
P
− be a canonical
symmetry, and let (
⋅
,
⋅
)
=
[
J
⋅
,
⋅
] be a canonical
scalar product. Note that one of these canonical objects uniquely determines
the others. Everywhere below, we fix, on ℋ, a unique form [
x
,
y
]
=
(
J
x
,
y
). At the same time, let us note that, in the question
we consider, a concrete choice of Hilbert scalar product is not really
essential. One needs only to fix the topology (defined by the above-mentioned
scalar product) and the structure of J. Let us mention the following result (see [15]) concerning some
redefinitions of a Hilbert structure in Krein spaces.Proposition 3.1.
Letℋ
=
ℋ
+
⊕
ℋ
− be a canonical
decomposition of the Krein space ℋ and let ℋ = ℒ
+
[
+
˙
]
ℒ
− be another
canonical decomposition of the same space. Let the first decomposition define
the canonical scalar product (
⋅
,
⋅
) and let the
second decomposition define the canonical scalar product (
⋅
,
⋅
)
1, that is, [
⋅
,
⋅
]
=
(
J
⋅
,
⋅
)
=
(
J
1
⋅
,
⋅
)
1. Then spectrum of the operator Q
=
J
1
J is strictly
positive, and the operator D
:
D
2
=
Q, σ
(
D
)
⊂
(
0
,
+
∞
), has the properties(3.1)
(a)
[
D
x
,
D
y
]
=
[
x
,
y
]
(b)
(
D
x
,
D
y
)
1
=
(
x
,
y
)
(c)
D
ℋ
+
=
ℒ
+
,
D
ℋ
−
=
ℒ
−
.
Definition 3.2.
The operatorD that we
introduced in Proposition 3.1 is called the canonical isometry that maps the Krein
space ℌ with the scalar
product (⋅
,
⋅) on the same
Krein space ℌ, but with the scalar product (
⋅
,
⋅
)
1. If it is necessary to exactly indicate the
corresponding scalar products or the canonical decompositions of the space ℌ, we replace D with(3.2)
D
ℌ
,
(
⋅
,
⋅
)
,
(
⋅
,
⋅
)
1
or
D
{
ℌ
+
,
ℌ
−
}
{
𝔏
+
,
𝔏
−
}
.Below nonnegative (especially maximal nonnegative) subspaces will play an
important role. The set of all maximal nonnegative subspaces of the Krein spaceℌ is denoted 𝔐
+
(
ℋ
).A subspaceℒ is called pseudoregular [16] if it can be presented in
the form(3.3)
ℒ
=
ℒ
^
+
˙
ℒ
1
,where ℒ
^ is a regular
subspace and ℒ
1 is an isotropic
part of ℒ (i.e., ℒ
1
=
ℒ
∩
ℒ
[
⊥
]).Proposition 3.3.
Let(i)
𝔏
+
∈
𝔐
+
(
ℌ
) and a
pseudoregular subspace,
(ii)
𝔏
0 the isotropic
subspace of 𝔏
+,
(iii)
(
⋅
,
⋅
)
′ a scalar
product on 𝔏
0 such that the
norm (
x
,
x
)
′ is equivalent
to the original one,
(iv)
𝔏
−
=
𝔏
+
[
⊥
],
and(3.4)
𝔏
+
=
𝔏
^
+
+
˙
𝔏
0
,
𝔏
−
=
𝔏
^
−
+
˙
𝔏
0
,where 𝔏
^
+ and 𝔏
^
− are uniformly
definite subspaces. Then one can define, on ℌ, a canonical scalar product (
⋅
,
⋅
) such
that(3.5)
(a)
on
𝔏
0
:
(
⋅
,
⋅
)
≡
(
⋅
,
⋅
)
′
(b)
𝔏
0
⊥
𝔏
^
+
,
𝔏
0
⊥
𝔏
^
−
(c)
on
𝔏
^
+
:
(
⋅
,
⋅
)
=
[
⋅
,
⋅
]
(d)
on
𝔏
^
−
:
(
⋅
,
⋅
)
=
−
[
⋅
,
⋅
]
.
Definition 3.4.
If a canonical scalar product
of Krein spaceℌ has (3.5), it
is said to be compatible with
(3.4) and the choice of the scalar product (
⋅
,
⋅
)
′ on 𝔏
1.
Definition 3.5.
Let𝔏 be a
pseudoregular subspace and let(3.6)
𝔏
=
𝔏
(
1
)
+
˙
𝔏
1
,
𝔏
=
𝔏
(
2
)
+
˙
𝔏
1be two different decompositions of 𝔏 as a direct sum
of a regular subspace and the isotropic subspace. We will call the map ℳ
𝔏
,
𝔏
(
1
)
,
𝔏
(
2
)
:
𝔏
(
1
)
↦
𝔏
(
2
), defined by the condition
(3.7)
[
ℳ
𝔏
,
𝔏
(
1
)
,
𝔏
(
2
)
x
,
y
]
=
[
x
,
y
]
(
∀
y
∈
𝔏
)
,
the standard map associated with (3.6).
Remark 3.6.
The mapℳ
𝔏
,
𝔏
(
1
)
,
𝔏
(
2
) is uniquely
determined by the condition (3.7) since the subspaces 𝔏
(
1
) and 𝔏
(
2
) are
projectionally complete and the subspace 𝔏
1 is neutral.
This map preserves the sesquilinear form [
⋅
,
⋅
] (i.e., is J-isometric).
Furthermore, the subspaces 𝔏
(
1
) and 𝔏
(
2
) can be
presented in the form(3.8)
𝔏
(
1
)
=
𝔏
+
(
1
)
[
+
˙
]
𝔏
−
(
1
)
,
𝔏
(
2
)
=
𝔏
+
(
2
)
[
+
˙
]
𝔏
−
(
2
)
,where 𝔏
+
(
1
) is uniformly
positive, 𝔏
−
(
1
) is uniformly
negative and(3.9)
𝔏
±
(
2
)
=
ℳ
𝔏
,
𝔏
(
1
)
,
𝔏
(
2
)
𝔏
±
(
1
)
.Then the union of (3.6) and (3.8) generates two decompositions of the
type (3.4).Define a special case of
pseudoregular subspaces: a nonnegative (non-positive) subspace ℒ is called a subspace of the class
h
+ (h
−) if it is
pseudoregular and dim
ℒ
1
<
∞ for ℒ
1 as in (3.3). In
Pontryagin spaces, every subspace is pseudoregular, and every semidefinite
subspace belongs to class h
+ or h
−.Here, the term
“operator” means “bounded linear operator.” By the symbolB
#, we denote the operator J-adjoint (J-a.) to an
operator B. Thus, if A
=
A
#, then A is a J-s.a. operator.
For an operator A, the symbol σ
(
A
) denotes its spectrum treated in the same way as in
[17] or [8].If an operator
family𝔜 is such that
the condition A
∈
𝔜 implies A
#
∈
𝔜, then this family is said to be J-symmetric.
An operator algebra 𝔄 is said to be a W
J
*-algebra if it
is closed in the weak operator topology, J-symmetric, and
contains the identity I. The symbol Alg
𝔜 means the
minimal W
J
*-algebra which
contains 𝔜.One of the most important directions in the
development of the operator theory is connected to the existence of invariant
maximal semidefinite subspaces for certain operator sets (see [18] for references) and the
study of the properties of the operators in such sets. A subspace ℒ is said to be A
-invariant (𝔜-invariant) if it is invariant with respect to the operator A (operator
family 𝔜).Proposition 3.7.
LetA be an operator
that acts in ℌ and has an
invariant pseudoregular subspace 𝔏, and let 𝔏
1 = 𝔏
∩
𝔏
[
⊥
] be also an A-invariant.
Consider two different decompositions of the space 𝔏,(3.10)
𝔏
=
𝔏
(
1
)
+
˙
𝔏
1
,
𝔏
=
𝔏
(
2
)
+
˙
𝔏
1
,as a direct sum of the isotropic part and a projectionally complete subspace.
Let(3.11)
A
|
𝔏
=
(
A
(
1
)
0
A
1
(
1
)
A
1
)
,
A
|
𝔏
=
(
A
(
2
)
0
A
1
(
2
)
A
1
)be two matrix representations of A
|
𝔏 corresponding
to these decompositions. Then(3.12)
A
(
1
)
=
(
ℳ
𝔏
,
𝔏
(
1
)
,
𝔏
(
2
)
)
A
(
2
)
(
ℳ
𝔏
,
𝔏
(
1
)
,
𝔏
(
2
)
)
−
1
,where ℳ
𝔏
,
𝔏
(
1
)
,
𝔏
(
2
) is the standard
map associated with (3.10).
Proposition 3.8.
Let(3.13)
ℌ
=
ℌ
+
[
+
˙
]
ℌ
−
,
ℌ
=
𝔏
+
[
+
˙
]
𝔏
−be two different canonical decompositions of the Krein space ℌ, and let ℌ
+ and 𝔏
+ be invariant
subspaces of a J-s.a. operator A. Then (3.14)
A
⋅
D
{
ℌ
+
,
ℌ
−
}
,
{
𝔏
+
,
𝔏
−
}
=
D
{
ℌ
+
,
ℌ
−
}
,
{
𝔏
+
,
𝔏
−
}
⋅
A
,where D
{
ℌ
+
,
ℌ
−
}
,
{
𝔏
+
,
𝔏
−
} are associated
with Definition 3.2.
Definition 3.9.
AJ-symmetric
operator family 𝔜 belongs to the
class D
κ
+ if there is a
subspace ℒ
+ in ℋ such that(i)
ℒ
+ is 𝔜-invariant,
(ii)
ℒ
+
∈
𝔐
+
(
ℋ
)
∩
h
+,
(iii)
dim
(
ℒ
+
∩
ℒ
+
[
⊥
]
)
=
κ.
Remark 3.10.
If aJ-symmetric
family 𝔜
∈
D
κ
+ and ℒ
+ is a 𝔜-invariant
subspace corresponding to Definition 3.9, then the pseudoregular subspace ℒ
+
[
⊥
] is 𝔜-invariant too.


## 3.2. Spectral Functions with Peculiarities

The spectral
resolution for different operator classes is one of the important problems in
the operator theory. Let us start with the following definition.LetΛ = {
λ
k
}
1
n be a finite set
of real numbers and let ℜ
Λ be the family {
X
} of all Borel
subsets of ℝ such that ∂
X
∩
Λ
=
∅, where ∂
X is the boundary
of X in ℝ. Let E
:
X
↦
E
(
X
) be a countably
additive (in the weak topology) function, that maps ℜ
Λ to a
commutative algebra of projections in a Hilbert space ℋ, where E
(
[
λ
−
ϵ
,
λ
+
ϵ
]
)
≠
0 for every λ
∈
Λ, and ϵ
>
0, and, moreover, E
(
ℝ
)
=
I. E
(
X
) is called a spectral function on
ℝ
with the
peculiar spectral set
Λ; the mention of Λ can be omitted.
The symbol Supp
(
E
) means the
minimal closed subset of ℝ such that E
(
X
)
=
0 for every X: X
⊂
ℝ
∖
S and X
∈
ℜ
Λ. Besides the symbol E, we will use also, as a notation of a spectral
function, the symbol E
λ, λ
∈
ℝ, where E
λ = E
(
(
−
∞
,
λ
)
).A spectral
functionE, that acts in a Krein space, is said to be J-orthogonal or J-s.a. if E
(
X
) is a J-orthoprojection
for every X
∈
ℜ
Λ.Let us recall the
definition ofscalar spectral
operator with real spectrum [19]. An operator A acting in a
Hilbert space is said to be a scalar spectral operator if there exists a
spectral function E with empty
peculiar spectral set Λ such that, for
every X
∈
ℜ
Λ
:
E
(
X
)
A
=
A
E
(
X
), σ
(
A
|
E
(
X
)
ℋ
)
⊂
X
¯ and A
E
(
X
) = ∫
X
ξ
E
(
d
ξ
) in the weak
sense.Now, letE be a spectral
function with peculiar spectral set Λ. A scalar function f
(
ξ
) is said to be defined almost everywhere (with respect to E), to have a finite value almost everywhere,
and so on, if the corresponding property holds almost everywhere in the weak
sense on an arbitrary set X
∈
ℜ
Λ, X
∩
Λ
=
∅. We will assume that the function f
(
ξ
) is not defined
at Λ.The following
theorem was announced in [2] and proved in [20].Theorem 3.11.
Let𝔜
∈
D
κ
+ be a
commutative family of J-s.a. operators
with real spectra. Then there exists a J-orthogonal
spectral function E
λ with a finite
number of spectral peculiarities Λ (Λ may be the
empty set) such that the following conditions hold(3.15)
(a)
E
λ
∈
Alg
𝔜
∀
λ
∈
ℝ
∖
Λ
;
(b)
there is a nonnegative subspace
ℒ
+
,
corresponding to Definition 3
.9
,
for which the descomposition
E
(
Δ
)
ℋ
=
E
(
Δ
)
ℒ
+
[
+
˙
]
E
(
Δ
)
ℒ
−
holds
,
Δ
being any closed segment
Δ
⊂
ℝ
satisfying
Δ
∈
ℜ
Λ
and
Δ
∩
Λ
=
∅
;
(c)
for every operator
A
∈
𝔜
,
there exists a defined almost everywhere
and (uniformly) bounded function
ϕ
(
λ
)
such that for every interval
Δ
⊂
ℝ
,
Δ
∈
ℜ
Λ
,
Δ
∩
Λ
=
∅
,
the decomposition
A
E
(
Δ
)
=
∫
Δ
ϕ
(
λ
)
E
(
d
λ
)
is valid
;
(d)
the subspace
ℋ
˜
=
CLin
Δ
∈
ℜ
Λ
,
Δ
∩
Λ
=
∅
{
E
(
Δ
)
ℋ
}
is pseudoregular
and its isotropic part has finite dimension
;
(e)
if
λ
0
∈
Λ
and
ℋ
λ
0
:
=
∩
λ
0
∈
Δ
E
(
Δ
)
ℋ
,
then for every operator
A
∈
𝔜
the set
σ
(
A
|
ℋ
λ
0
)
is as ingletone
{
μ
A
}
and there is a positive integer
m
A
such
that
(
A
−
μ
A
I
)
m
A
|
ℋ
λ
0
=
0
;
(f)
if
λ
0
∈
Λ
,
then
limsup
⁡
λ
→
λ
0
∥
E
λ
∥
=
∞
or at least for one
A
∈
𝔜
the operator
A
|
ℋ
λ
0
is not a spectral operator of scalar type
.A
spectral function E with a peculiar
spectral set Λ, satisfying ((3.15)), is called an eigen spectral function (e.s.f.) of the
operator family 𝔜.Definition 3.12.
LetE
λ be an e.s.f. of
an operator family 𝔜 and let an
operator A
∈
𝔜 and a function ϕ
(
λ
) be connected by
the system of equalities from (3.15)(c). Then the function ϕ
(
λ
) is said to be the portrait of the operator A, and the operator A is said to be the original of ϕ
(
λ
) in 𝔜 (with respect
to E
λ).Let a spectral function E with a peculiar
spectral set Λ be an e.s.f. of 𝔜. If λ
∈
Λ, then λ will be called
a peculiarity of 𝔜. Let λ be a
peculiarity. Fix a set X
∈
ℜ
Λ: X
∩
Λ
=
{
λ
}. The peculiarity λ is called regular if the operator family {
E
(
X
∩
Y
)
}
Y
∈
ℜ
Λ is bounded;
otherwise, it is called singular.
Note that the notion of regular and singular peculiarities is correctly defined
since the boundedness of the family {
E
(
X
∩
Y
)
}
Y
∈
ℜ
Λ does not depend
on X.


## 3.3. Some Function Spaces

Assume thatσ
(
t
) is a
nondecreasing function defined on the segment [
−
1
;
1
], continuous in the points −
1, 0, and 1, continuous (at least) from the left in all other
points of the segment, and having an infinite number of growth points, where
zero is one of these points. This function σ
(
t
) generates, on [
−
1
;
1
], a Lebesgue-Stieltjes measure μ
σ and spaces (L
σ
2, L
σ
∞, etc.) of complex-valued functions. At the same time
we will consider also some spaces of vector-valued functions; so, from time to
time, we will note, after a symbol of a space a symbol of a range for the
functions forming this space, for instance, L
σ
2
(
ℂ
). Next, let us consider a slightly different
construction. Let G
(
t
) be a μ
σ-measurable
function defined a.e. on [
−
1
;
1
] and such that
a.e. G
(
t
)
≥
1 for every τ
∈
(
0
;
1
], it is true that ∫
−
1
−
τ
G
(
t
)
d
σ
(
t
)
<
∞ and ∫
τ
1
G
(
t
)
d
σ
(
t
)
<
∞, but ∫
−
1
1
G
(
t
)
d
σ
(
t
) = ∞. Set(3.16)
ν
(
τ
)
=
{
∫
−
1
τ
G
(
t
)
d
σ
(
t
)
,
if
τ
∈
[
−
1
;
0
]
;
−
∫
τ
1
G
(
t
)
d
σ
(
t
)
,
if
τ
∈
[
0
;
1
]
;
The function ν
(
t
) is
nondecreasing in both segments [
−
1
;
0
) and (
0
;
1
], but it is unbounded in neighborhoods of zero.
Define, for it, a corresponding function space. Let f
(
t
) and g
(
t
) be arbitrary
functions continuous in [
−
1
;
1
] and vanishing
in some neighborhoods (different in the general case for f
(
t
) and g
(
t
)) of zero. Then
the integral ∫
−
1
1
f
(
t
)
g
(
t
)
¯
d
ν
(
t
) is well defined
and generates a structure of pre-Hilbert space on the set of all such
functions. The completion of the space will be denoted L
ν
2 (or L
ν
2
(
ℂ
)). Note that,
due to (3.16), the spaces L
σ
∞ and L
ν
2 form a Banach
pair, so the space L
σ
∞
∩
L
ν
2 is well defined
(for details, see [21]).Let us pass to
some notation relating to direct integrals of Hilbert spaces and corresponding
model descriptions of self-adjoint operators (see [22, Section 41]; [23, Chapter 7]; [24, Chapter 4.4]; [25, Chapter VII]). We will use
definitions close to the “coordinate notation” given in [22]. A difference between
[22] and the
definitions that follow is related to the fact that direct integrals, here,
will be used not only for a resolution of Hilbert spaces but also for a
resolution of Krein spaces. Let ℰ be some
separable Hilbert space (ℰ can be finite
dimensional), let {
d
j
}
1
α be an
orthonormalized basis of this space, let σ
(
t
) be be the same
as above. Let {
ρ
j
(
t
)
}
1
α be a system of
nonnegative μ
σ-measurable
functions defined almost everywhere (a.e) on the segment [
−
1
;
1
] and such that
every function of the system is the indicator of some set of nonzero measure
and μ
σ
{
t
:
ρ
j
(
t
)
=
0
,
j
=
1
,
2
,
…
,
α
}
=
0. Denote(3.17)
d
σ
→
(
t
)
=
∑
j
=
1
α
d
j
ρ
j
(
t
)
d
σ
(
t
)
.In this case the sum in the right part of the formula is considered as a
formal expression without any suggestion of its convergence or divergence.Here the spaceM
σ
→
(
ℰ
) means the space
of vector-valued functions {
f
(
t
)
} defined a.e.
(with respect to μ
σ) on the
segment [
−
1
;
1
] and taking
values in ℰ under the
conditions(3.18)
f
(
t
)
=
∑
j
=
1
α
β
j
(
t
)
d
j
,where β
j
(
t
) runs the set of
all μ
σ-measurable
a.e. finite scalar functions such that(3.19)
(a)
β
j
(
t
)
=
ρ
j
(
t
)
β
j
(
t
)
,
j
=
1
,
2
,
…
,
α
;
(b)
a
.e
.
∥
f
(
t
)
∥
ℰ
2
=
∑
j
=
1
α
|
β
j
(
t
)
|
2
<
∞
.The topology on M
σ
→
(
ℰ
) is introduced
by a base for neighborhoods of zero, where any neighborhood of the base is
defined by a couple of positive numbers ϵ and δ (the couples
are different for the different neighborhoods) and contains all functions
satisfying the condition μ
σ
{
t
:
∥
f
(
t
)
∥
ℰ
2
≥
δ
}
<
ϵ. Next, the symbol L
σ
→
2
(
ℰ
) means here a
Hilbert space of functions f
(
t
)
∈
M
σ
→
(
ℰ
) such that ∫
−
1
1
∥
f
(
t
)
∥
ℰ
2
d
σ
(
t
)
<
∞.The spacesM
σ
→
(
ℰ
) and L
σ
→
2
(
ℰ
) are said to be
a standard space of measurable
functions and a standard Hilbert space, respectively.The choice of{
d
j
}
1
α in the
construction of spaces M
σ
→
(
ℰ
) and L
σ
→
2
(
ℰ
) is not essential
if(3.20)
ρ
j
(
t
)
=
ρ
l
(
t
)
for
l
,
j
=
1
,
2
,
…
,
α
.Spaces L
σ
→
2
(
ℰ
) of this type
correspond, in particular, to model representations of self-adjoint operators
with uniform multiplicity (see [25]). So M
σ
→
(
ℰ
) and L
σ
→
2
(
ℰ
) are said to be spaces of uniform multiplicity
α if conditions
(3.20) are fulfilled.IfL
σ
→
2
(
ℰ
) is not a space
of uniform multiplicity, it can be represented as an orthogonal sum of spaces
of uniform multiplicity. Note a special case of such representation.Definition 3.13.
A
spaceL
σ
→
2
(
ℰ
) is said to be
orderly decomposable on uniform components if (see (3.17))(3.21)
L
σ
→
2
(
ℰ
)
=
L
σ
→
1
2
(
ℰ
1
)
⊕
L
σ
→
2
2
(
ℰ
2
)
⊕
…
,where ℰ
=
⊕
∑
j
=
1
β
ℰ
j (both β
=
∞ and β
<
∞ can occur), the
above decomposition of ℰ is concordant
with the choice of the basis {
d
j
}
1
α in the sense
that ℰ
j
=
CLin
{
d
j
}
d
j
∈
ℰ
j, L
σ
→
j
2
(
ℰ
j
) is a space of
uniform multiplicity dim
ℰ
j, ℰ
j
⊂
ℰ
j
+
1, dim
ℰ
j
<
dim
ℰ
j
+
1
,
σ
j
(
t
) = ∫
−
1
t
χ
j
(
τ
)
d
σ
(
τ
), χ
j
(
τ
) is the
indicator of some μ
σ-measurable set X
j
,
μ
X
j
≠
0, d
σ
→
j
(
t
)
=
(
∑
l
:
d
l
∈
ℰ
j
d
l
)
d
σ
→
j
(
t
), X
j
∩
X
l
=
∅, j, l
=
1
,
2
,
…
,
β, j
≠
l.Spaces orderly
decomposable on uniform components play the key role in the theory of model
representation for self-adjoint operators (for details, see, [25, Theorem VII.6]). Practically
the same definition can be given for spaces M
σ
→
(
ℰ
).
Definition 3.14.
M
σ
→
(
ℰ
) is said to be orderly decomposable on uniform components if(3.22)
M
σ
→
(
ℰ
)
=
M
σ
→
1
(
ℰ
1
)
+
M
σ
→
2
(
ℰ
2
)
+
…
,where M
σ
→
j
(
ℰ
j
) is a space of
uniform multiplicity, the rest of the elements in (3.22) are the same as in
(3.21).We introduce some notation related to
multiplication operators by scalar functions. Everywhere below we assume a
scalar function φ
(
t
) to be defined
a.e. on [
−
1
;
1
], μ
σ-measurable,
and a.e. bounded. For f
(
t
)
∈
M
σ
→
(
ℰ
), set(3.23)
(
Φ
f
)
(
t
)
=
φ
(
t
)
f
(
t
)
.It is clear that (
Φ
f
)
(
t
)
∈
M
σ
→
(
ℰ
), so (3.23) defines on M
σ
→
(
ℰ
) the continuous
operator Φ (= the
multiplication operator by the function φ
(
t
)). If φ
(
t
) satisfies some
additional conditions, one can consider the operator Φ as acting
simultaneously on different spaces. If, for instance, φ
(
t
) is continuous,
then the operator Φ is well defined
on every space M
σ
(
ℰ
) independently
of σ
→
(
t
) and ℰ. If φ
(
t
)
∈
L
σ
∞
(
ℂ
), then L
σ
→
2
(
ℰ
) can also be
taken as a domain of Φ. So, if it is necessary, we will mention,
simultaneously, the operator Φ and its domain
using the notation {
Φ
,
𝔇
(
Φ
)
}, say, {
Φ
,
L
σ
→
2
(
ℰ
)
}.LetX
τ be the
multiplication operator by the indicator χ
[
−
1
;
τ
)
(
t
) of the set [
−
1
;
τ
), X
τ
=
{
X
τ, L
σ
→
2
(
ℰ
)
}. Pass to the description of automorphisms acting on L
σ
→
2
(
ℰ
) and commuting
with operators X
τ, τ
∈
[
−
1
;
1
]. Denote, by 𝒢
t, the subspace of the space ℰ spanned by all
vectors of the set {
d
j
}
j
=
1
α such that ρ
j
(
t
)
≠
0 (see (3.17)).
The following result is well known (e.g., Birman, Solomjak [23, Chapter 7]; [24, Theorem 4.4.6]; [22, Proposition 1, Subsection
2; Section 41]).Proposition 3.15.
Let𝒰 be a unitary
operator commuting with {
X
τ
,
L
σ
→
2
(
ℰ
)
} for a.a. τ
∈
[
−
1
;
1
]. Then there is an operator-valued weakly μ
σ-measurable
function U
t defined a.e. on [
−
1
;
1
] such that, for
a.e. t
∈
[
−
1
;
1
], the operator U
t is unitary on ℰ and(3.24)
(a)
(
𝒰
f
)
(
t
)
=
U
t
f
(
t
)
;
(b)
U
t
𝒢
t
=
𝒢
t
;
(c)
U
t
|
𝒢
t
⊥
=
I
|
𝒢
t
⊥
.
Remark 3.16.
The spaceL
σ
→
2
(
ℰ
) is a complete
linear set in M
σ
→
(
ℰ
). So if initially an operator 𝒰 acts on L
σ
→
2
(
ℰ
) and satisfies
conditions (3.24), its domain can be extended to M
σ
→
(
ℰ
) via the passage
to the limit. Thus the operator {
𝒰
,
M
σ
→
(
ℰ
)
} is a continuous
bijective mapping.


## 4. On a Model Representation forJ-Orthogonal Spectral Functions without Peculiarities

We define, onL
σ
→
2
(
ℰ
), an additional structure of a Krein space. Let 𝒥 be an operator
acting on L
σ
→
2
(
ℰ
), being at the same time self-adjoint, unitary and
commuting with X
τ for a.e. τ
∈
[
−
1
;
1
]. By Proposition 3.15, the operator 𝒥 has the
representation(4.1)
(
𝒥
f
)
(
t
)
=
J
t
f
(
t
)
,
t
∈
[
−
1
;
1
]
,where J
t is self-adjoint
a.e. on [
−
1
;
1
]. Let(4.2)
[
f
1
(
t
)
,
f
2
(
t
)
]
ℰ
=
(
J
t
f
1
(
t
)
,
f
2
(
t
)
)
ℰ
.Then the inner product(4.3)
[
f
1
(
t
)
,
f
2
(
t
)
]
L
σ
→
2
(
ℰ
)
=
∫
−
1
1
[
f
1
(
t
)
,
f
2
(
t
)
]
ℰ
d
σ
(
t
)
.converts the space L
σ
→
2
(
ℰ
) to a Krein
space. This Krein space is denoted 𝒥
−
L
σ
→
2
(
ℰ
) and it is said
to be a standard Krein space.Remark 4.1.
The product[
f
1
(
t
)
,
f
2
(
t
)
]
ℰ is a.e. well
defined both for f
1
(
t
)
,
f
2
(
t
)
∈
L
σ
→
2
(
ℰ
) and f
1
(
t
)
,
f
2
(
t
)
∈
M
σ
→
(
ℰ
).A standard
Krein space can be used for a model representation of aJ-spectral
function (= J-orth.sp.f.) E
λ without
peculiarities. For simplicity everywhere below, we will assume
that(4.4)
E
−
1
=
0
,
E
+
1
=
I
,
E
−
1
=
E
−
1
+
0
.
Definition 4.2.
LetE
λ be a J-orth.sp.f.
with the empty set of peculiarities. A space 𝒥
−
L
σ
→
2
(
ℰ
) is said to be a model
space for E
λ if, for some
canonical scalar product on ℌ, there is an isometric and J-isometric
operator W
:
𝒥
−
L
σ
→
2
(
ℰ
)
↦
ℌ such that, for
every λ
∈
[
−
1
;
1
],(4.5)
E
λ
=
W
X
λ
W
−
1
,where, as above, X
λ is the
multiplication operator by the indicator χ
[
−
1
;
λ
)
(
t
) of the set [
−
1
;
λ
), X
λ
=
{
X
λ
,
𝒥
−
L
σ
→
2
(
ℰ
)
}. The operator W is said to be
an operator
of similarity.Proposition 4.3.
EveryJ-orth.sp.f. E
λ with an empty set
of peculiarities has a model space 𝒥
−
L
σ
→
2
(
ℰ
).
Proof.
By
[19, Lemma XV.6.2],
one can define, on ℋ, a new scalar product (noncanonical in the general
case) (
⋅
,
⋅
)
1, topologically equivalent to the initial scalar product (
⋅
,
⋅
) and such that
the spectral function E
λ, is orthogonal with respect to (
⋅
,
⋅
)
1. Then [
⋅
,
⋅
]
=
(
G
⋅
,
⋅
)
1, where the operators E
λ commute with G for a.a. λ
∈
[
−
1
;
1
]. Now one can introduce, on ℋ, a new canonical scalar product (
⋅
,
⋅
)
2
=
(
|
G
|
⋅
,
⋅
)
1, where |
G
|
=
(
G
2
)
1
/
2 is the module
of G. The new scalar product generates the canonical
decomposition ℋ
=
ℋ
+
(
2
)
⊕
2
ℋ
−
(
2
). It easy to show that the subspaces ℋ
+
(
2
) and ℋ
−
(
2
) are E
λ-invariant for
a.a. λ
∈
[
−
1
;
1
]. Thus there are model spaces 𝒥
−
L
σ
→
+
2
(
ℰ
+
) and 𝒥
−
L
σ
→
−
2
(
ℰ
−
) for orth.sp.f. E
λ
|
ℋ
+
(
2
) and E
λ
|
ℋ
−
(
2
), respectively. Without loss of generality one can
assume that the scalar functions (see (3.17)) σ
+
(
t
) and σ
−
(
t
) are such that(4.6)
σ
+
(
t
)
=
∫
−
1
t
ρ
+
(
λ
)
d
σ
(
λ
)
,
σ
−
(
t
)
=
∫
−
1
t
ρ
−
(
λ
)
d
σ
(
λ
)
,
ρ
+
2
(
λ
)
=
ρ
+
(
λ
)
,
ρ
−
2
(
λ
)
=
ρ
−
(
λ
)
,
σ
(
t
)
=
∫
−
1
t
(
ρ
+
(
λ
)
+
ρ
−
(
λ
)
−
ρ
+
(
λ
)
ρ
−
(
λ
)
)
d
σ
(
λ
)
.
Put 𝒥
−
L
σ
→
2
(
ℰ
)
:
=
𝒥
−
L
σ
→
+
2
(
ℰ
+
)
⊕
𝒥
−
L
σ
→
−
2
(
ℰ
−
). The rest is straightforward.
Remark 4.4.
As is well known, a model
representation for an orth.sp.f. is not uniquely determined. At the same time,
all scalar measures for such representations in the case of the same orth.sp.f.
are equivalent among themselves and the function (see (3.24)) r
(
t
)
=
dim
𝒢
t does not depend
on the choice of the model representation. This is the reason why a model
representation for orth.sp.f., usually is taken in the class of Lebesgue spaces
orderly decomposable on uniform components. In the case of J-orth.sp.f.,
the situation is slightly more complicated. Specifically, for all model
representations for a fixed J-orth.sp.f.,
all admissible scalar measures are equivalent among themselves and two
functions r
+
(
t
)
=
dim
Ker
(
(
J
t
−
I
)
|
𝒢
t
) and r
−
(
t
)
=
dim
Ker
(
(
J
t
+
I
)
|
𝒢
t
) do not depend
among themselves and of the choice on the model representation. Conversely, if
data include a class of equivalent measures, functions r
+
(
t
) and r
−
(
t
), then a J-orth.sp.f.
with empty set of peculiarities can be restored up to J-isometric
equivalence. In particular, one can choose a suitable function σ
(
t
), construct a space 𝒥
−
L
σ
→
+
2
(
ℰ
+
) using σ
(
t
) and r
+
(
t
), construct a space 𝒥
−
L
σ
→
−
2
(
ℰ
−
) using σ
(
t
) and r
−
(
t
), and put 𝒥
−
L
σ
→
2
(
ℰ
)
:
=
𝒥
−
L
σ
→
+
2
(
ℰ
+
)
⊕
𝒥
−
L
σ
→
−
2
(
ℰ
−
). Note that, even if the spaces 𝒥
−
L
σ
→
+
2
(
ℰ
+
) and 𝒥
−
L
σ
→
−
2
(
ℰ
−
) are orderly
decomposable on uniform components, the spaces 𝒥
−
L
σ
→
2
(
ℰ
) in the general
case do not have this property.
Proposition 4.5.
Assume
thatE
λ is a J-orth.sp.f.
with empty set of peculiarities, 𝒥
−
L
σ
→
2
(
ℰ
) and 𝒥
′
−
L
σ
→
2
(
ℰ
) are two
different standard Krein spaces such that each of them is a model space for E
λ and is derived
from the same space L
σ
→
2
(
ℰ
) by canonical
symmetries 𝒥 and 𝒥
′, respectively. Then, in L
σ
→
2
(
ℰ
), there is a unitary operator 𝒰 satisfying
(3.24) and such that 𝒥
′
=
𝒰
−
1
𝒥
𝒰.
Proof.
In the
general case the operators of similarityW and W
′ correspond to
different canonical scalar products on the same space ℋ, say, W
:
𝒥
−
L
σ
→
2
(
ℰ
)
↦
{
ℋ
,
(
⋅
,
⋅
)
} and W
′
:
𝒥
′
−
L
σ
→
2
(
ℰ
)
↦
{
ℋ
,
(
⋅
,
⋅
)
1
}, where (
⋅
,
⋅
)
:
=
[
J
⋅
,
⋅
] and (
⋅
,
⋅
)
1
:
=
[
J
1
⋅
,
⋅
]. On the other hand thanks to Definition 4.2, E
λ is self-adjoint
simultaneously with respect to the scalar products (
⋅
,
⋅
) and (
⋅
,
⋅
)
1, hence it commutes with J and J
1. Thus, taking into account Proposition 3.1, we can assume
that (
⋅
,
⋅
)
=
(
⋅
,
⋅
)
1. Then the required operator 𝒰 can be defined
by the formula 𝒰
:
=
W
−
1
W
′. Indeed, first, this 𝒰 is unitary
because the canonical scalar product on ℋ is the same for W and W
′; second, for every function f
(
t
)
∈
L
σ
→
2
(
ℰ
) is true, (
𝒥
′
f
(
t
)
,
f
(
t
)
)
L
σ
→
2
(
ℰ
)
=
[
f
(
t
)
,
f
(
t
)
]
𝒥
′
−
L
σ
→
2
(
ℰ
)
=
[
𝒰
f
(
t
)
,
𝒰
f
(
t
)
]
𝒥
−
L
σ
→
2
(
ℰ
)
=
[
𝒥
𝒰
f
(
t
)
,
𝒰
f
(
t
)
]
L
σ
→
2
(
ℰ
); third, 𝒰
X
τ
f
(
t
)
=
W
−
1
E
τ
W
′
f
(
t
)
=
X
τ
𝒰
f
(
t
). The rest follows from Proposition 3.15.The proof of the next proposition will be omitted because it
contains the same well-known ideas as the proof of Proposition
3.15.Proposition 4.6.
Letℳ be a J-unitary
operator in a standard Krein space 𝒥
−
L
σ
→
2
(
ℰ
) commuting with {
X
τ
,
𝒥
−
L
σ
→
2
(
ℰ
)
} for a.a. τ
∈
[
−
1
;
1
]. Then there is an operator-valued weakly measurable
function M
t defined a.e. on [
−
1
;
1
] and taking
values in the set of bounded operators acting in ℰ such that, for
a.a. t
∈
[
−
1
;
1
],(4.7)
(a)
(
ℳ
f
)
(
t
)
=
M
t
f
(
t
)
;
(b)
M
t
𝒢
t
=
𝒢
t
,
M
t
𝒢
t
⊥
=
𝒢
t
⊥
,
M
t
|
𝒢
t
⊥
=
I
|
𝒢
t
⊥
;
(c)
[
M
t
x
,
M
t
x
]
=
[
x
,
x
]
∀
x
∈
ℰ
;
(d)
EssSup
t
∈
[
−
1
;
1
]
∥
M
t
∥
ℰ
=
∥
ℳ
∥
L
σ
→
2
(
ℰ
)
.


## 5. Unbounded Elements in Hilbert Spaces

First, let us
consider the notion of unbounded elements.Assume thatℋ is a Hilbert
space, P
t is a resolution
of the identity (= an orthogonal spectral function with the empty set of peculiar
points) defined on the segment [
−
1
;
1
], continuous in zero (with respect to the w-topology),
and(5.1)
(a)
P
−
1
=
0
,
P
1
=
I
;
(b)
for every
t
∈
[
−
1
;
1
]
the unilateral limits
w
−
lim
μ
→
t
−
0
P
μ
and
w
−
lim
⁡
μ
→
t
+
0
P
μ
exist, where for definiteness
P
t
−
0
=
P
t
.Set P
λ
,
μ
=
I
+
P
λ
−
P
μ
+
0, where λ
∈
[
−
1
;
0
), μ
∈
(
0
;
1
].Next, letx
λ
,
μ be a mapping of
the numerical set [
−
1
;
0
)
×
(
0
;
1
] into ℋ (λ
∈
[
−
1
;
0
), μ
∈
(
0
;
1
]). The function x
λ
,
μ is said to be conformed with P
t if the
following condition is fulfilled; for every λ
,
α
∈
[
−
1
;
0
), μ
,
β
∈
(
0
;
1
]; the equality P
λ
,
μ
x
α
,
β
=
x
γ
,
δ holds, where γ
=
min
⁡
{
λ
,
α
}, δ
=
max
⁡
{
μ
,
β
}.Note thatx
λ
,
μ has the
following property:(5.2)
if
sup
λ
∈
[
−
1
;
0
)
μ
∈
(
0
;
1
]
⁡
{
∥
x
λ
,
μ
∥
}
<
∞
then there is an element
x
∈
ℋ
such that for every
λ
∈
[
−
1
;
0
)
,
μ
∈
(
0
;
1
]
the equality
x
λ
,
μ
=
P
λ
,
μ
x
holds
.It is clear that the element x from (5.2) is
uniquely defined by x
λ
,
μ and can be
found by the formula x
=
w
−
lim
⁡
λ
→
−
0
μ
→
+
0
x
λ
,
μ.Definition 5.1.
A functionx
λ
,
μ, which is conformed with P
λ, is said to be an unbounded element conformed with
P
λ (or, if it
cannot produce a misunderstanding or an unbounded element) if sup
λ
∈
[
−
1
;
0
)
μ
∈
(
0
;
1
]
{
∥
x
λ
,
μ
∥
}
=
∞.Note that unbounded
elements conformed with P
t exist if and
only if zero is a point of growth for P
t, that is, for every ϵ
> 0 it is true
that P
+
ϵ
−
P
−
ϵ
≠
0. Everywhere below in this section, this condition for P
t is assumed to
be fulfilled.For brevity
everywhere below unbounded elements will be denoted by symbolsx
˜, y
˜, and so on. For λ
∈
[
−
1
;
0
), μ
∈
(
0
;
1
], we set x
λ
,
μ
:
=
P
λ
,
μ
x
˜.Definition 5.2.
Unbounded elementsx
˜
1
,
x
˜
2
,
…
,
x
˜
k, conformed with a (common) resolution of the identity P
λ, are said to be linearly independent modulo
ℋ if every
nontrivial linear combination of them is an unbounded element from ℋ.Next, for every
bounded P
t-measurable
function ϕ
(
t
), one can define, on ℋ, the following operator Φ:(5.3)
Φ
x
:
=
∫
−
1
1
ϕ
(
t
)
d
P
t
x
.Using the previous notation, rewrite the last formula as(5.4)
(
Φ
x
˜
)
λ
,
μ
=
∫
−
1
λ
ϕ
(
t
)
d
P
t
x
˜
+
∫
μ
1
ϕ
(
t
)
d
P
t
x
˜
.Representation (5.4) allows a possibility to treat the operator Φ in a more
general sense; unbounded elements conformed with P
t can be
naturally included to the domain of Φ. Note that the portrait of an unbounded element can
be both a bounded element and an unbounded element. Moreover, ϕ
(
t
) in (5.4) could
also be taken unbounded. In this case the portrait of a bounded element,
optionally, is an unbounded element.Proposition 5.3.
If a functionϕ
(
t
) is unbounded on
every subset of complete E
t-measure, then
the vector space Φ
ℋ contains an
infinite number of unbounded elements conformed with P
t and a linearly
independent modulo ℋ.The last proposition
shows (see also below, Proposition 5.4) that if we need to operate with a
finite number of unbounded elements, then only bounded functions ϕ
(
t
) are admissible
in (5.4).Until the end
of the present section, the orthogonal resolution of the identityP
t and a
nondecreasing function σ
(
t
), such that μ
σ-measurability
on [
−
1
;
1
] coincides with
the P
t-measurability,
will be fixed. Note that the existence of σ
(
t
) follows from
the separability of ℋ (see [26, Section 76, Theorem 1]).We give an
additional notation. Let{
x
˜
j
}
1
k be a fixed
family of unbounded elements conformed with P
t and a linearly
independent modulo ℋ. Both the unbounded elements and the ordinary vectors
from ℋ can be
considered as functions defined on [
−
1
;
0
)
×
(
0
;
1
] and taking
values in ℋ (see (5.2)).
The linear span of vectors from ℋ and unbounded
elements from {
x
˜
j
}
1
k, consistently taken as functions on [
−
1
;
0
)
×
(
;
1
], is denoted ℋ
˜. Additionally ℋ
˜ will be
considered as a Hilbert space, where ℋ is a subspace
with the the same scalar product that was given on ℋ from the
beginning, and unbounded elements from {
x
˜
j
}
1
k are mutually
orthogonal and orthogonal to ℋ. The space ℋ
˜ is said to be
an expansion of ℋ (generated by {
x
˜
j
}
1
k).Next, using the
system{
x
˜
j
}
1
k, introduce the function ν
(
t
);(5.5)
ν
(
t
)
=
{
σ
(
t
)
+
∑
j
=
1
k
∥
P
t
x
˜
j
∥
2
,
if
t
∈
[
−
1
,
0
)
;
−
σ
(
1
)
+
σ
(
t
)
−
∑
j
=
1
k
∥
(
P
t
−
I
)
x
˜
j
∥
2
,
if
t
∈
[
0
,
1
)
.
The connection between P
t and σ
(
t
) implies that
the function ν
(
t
) introduced in
(5.5) has (3.16). In this case the function G
(
t
) from (3.16) can
be calculated directly through ν
(
t
) and σ
(
t
).Proposition 5.4.
Let{
x
˜
j
}
1
k be a system of
unbounded elements, forming together with ℋ the space ℋ
˜, let ϕ
(
t
) be a μ
σ-measurable
function, let Φ be the operator
defined by (5.4). Then Φ
ℋ
˜
⊂
ℋ if and only if ϕ
(
t
)
∈
L
σ
∞
∩
L
ν
2.
Proof.
Sufficiency of the formulated condition is clear, so we will consider its
necessity only. Note that, by virtue of Proposition5.3, condition Φ
ℋ
˜
⊂
ℋ implies the
boundedness of the function ϕ
(
t
). Next, since Φ
x
˜
j
∈
ℋ for j
=
1
,
2
,
…
,
k, then, by (5.4) it is true that ∥
Φ
x
˜
j
∥
2
=
∫
−
1
1
|
ϕ
(
t
)
|
2
d
∥
P
t
x
˜
j
∥
2
<
∞ and, moreover, ∫
−
1
1
|
ϕ
(
t
)
|
2
d
σ
(
t
)
<
∞. Thus (5.5) implies ϕ
(
t
)
∈
L
σ
∞
∩
L
ν
2.For future
applications, both the casesΦ
ℋ
˜
⊂
ℋ and(5.6)
Φ
ℋ
˜
⊂
ℋ
˜are important.Proposition 5.5.
Let{
x
˜
j
}
1
k be a system of
unbounded elements, generating together with ℋ the space ℋ
˜. Then there are no more than k
2
μ
σ-measurable
functions ϕ
(
t
) linearly
independent modulo L
σ
∞
∩
L
ν
2 such that (5.6)
is fulfilled.
Proof.
Due to
(5.6), every element from the system {
Φ
x
˜
j
}
1
k can be uniquely
represented (modulo ℋ) as a linear
combination of elements from {
x
˜
j
}
1
k, that is,(5.7)
Φ
x
˜
j
=
∑
m
=
1
k
c
j
m
x
˜
m
(
mod
⁡
ℋ
)
,
j
=
1
,
2
,
…
,
k
.Thus the operator Φ generates the
matrix(5.8)
C
Φ
=
(
c
j
m
)
j
,
m
=
1
k
,in addition, Φ
ℋ
˜
⊂
ℋ if and only if C
Φ
=
0. Hence functions comparable modulo L
σ
∞
∩
L
ν
2 have the same
matrix (5.8).Remark 5.6.
For the functionϕ
(
t
)
≡
1, (5.7) is always fulfilled. At the same time the
example of the unbounded elements {
(
1
)
m
}
m
=
1
∞ and {
(
i
)
m
}
m
=
1
∞ from the space l
2 shows that the case, where there are no functions
different from a constant modulo L
σ
∞
∩
L
ν
2 and simultaneously
satisfying (5.6), is possible. Moreover, the estimation mentioned in
Proposition 5.5 is excessive for every k
>
1 because two
arbitrary matrices of the form (3.19) generated by (5.7) commute. For a
discursion of the linear dimension of a group of commutative matrices, see, for
instance, [27, Part
2, Section 10].


## 6. A Function Model for aJ-Symmetric Family of the Class D
κ
+

## 6.1. Some Remarks

In this section
a function model ofJ-symmetric
family 𝔜
∈
D
κ
+ with real
spectrum will be discussed. This model is defined with the help of an e.s.f. E
λ of 𝔜 (see Section
3). It is incomplete because the model describes the
operator family restricted on some important subspaces (in particular, on the subspace ℋ
˜ (see
(3.15)(d))) and not this family itself. By virtue of Theorem 3.11, it is clear that the general situation
can be reduced to the case of J-orth.sp.f. E
λ with a unique
spectral peculiarity in zero. Furthermore, the case of a regular peculiarity is
trivial because, under this condition, all operators from 𝔜 are spectral in
the sense of Dunford and have a finite-dimensional nilpotent part. Thus E
λ is such
that(6.1)
(a)
E
−
1
=
E
−
1
+
0
=
0
,
E
+
1
=
I
;
(b)
Λ
=
{
0
}
;
(c)
sup
⁡
λ
∈
[
−
1
;
1
]
∖
{
0
}
{
∥
E
λ
∥
}
=
∞
.During the first stage, assume
additionally that, for every closed interval Δ
⊂
[
−
1
;
0
)
∪
(
0
;
1
], the following condition holds:(6.2)
if
E
(
Δ
)
≠
0,
then the subspace
E
(
Δ
)
ℋ
is positive
.Note that (6.2) does not
necessarily mean that ℋ is a Pontryagin
space because the subspace ℋ
˜
[
⊥
] can contain
both positive and negative subspaces of infinite dimension. We introduce some
notation. Let(6.3)
ℋ
1
=
ℋ
˜
∩
ℋ
˜
[
⊥
]
,
ℋ
2
=
ℋ
1
⊥
∩
ℋ
˜
,
ℋ
0
=
J
ℋ
1
,
P
j
be an
orthoprojection (in the sense of Hilbert spaces) onto
ℋ
j
,
j
=
0
,
1
,
2
,
E
˜
λ
:
=
E
λ
|
ℋ
˜
.Note that, by virtue of (6.1)(c),
the equality ℋ
1
≠
{
0
} holds, and due
to (6.2), the subspace ℋ
2 is uniformly
positive (recall that the subspace ℋ
1 is finite
dimensional). Without loss of generality one can assume that, on ℋ
2,(6.4)
(
⋅
,
⋅
)
=
[
⋅
,
⋅
]
.Now, we pass to a detailed analysis of the structure of E
λ. First, let X
∈
ℜ
Λ and(6.5)
X
⊂
[
−
1
;
1
]
∖
{
0
}
.Then(6.6)
E
(
X
)
|
(
ℋ
0
+
˙
ℋ
1
+
˙
ℋ
2
)
[
⊥
]
=
0
,thus it is enough to study the behavior of E
(
X
) on the
subspace(6.7)
ℋ
0
+
˙
ℋ
1
+
˙
ℋ
2
.Assume that a canonical scalar
product on ℋ is such that (
ℋ
0
+
˙
ℋ
1
+
˙
ℋ
2
)
[
⊥
]
⊥
(
ℋ
0
+
˙
ℋ
1
+
˙
ℋ
2
). In this case, (6.7) is an invariant subspace for J.A direct
verification shows that, if (6.5) holds, the operator E
(
X
) has the
following matrix realization:(6.8)
E
(
X
)
|
ℋ
0
+
˙
ℋ
1
+
˙
ℋ
2
=
(
0
0
0
E
10
(
X
)
0
E
12
(
X
)
E
20
(
X
)
0
E
22
(
X
)
)
,with respect to (6.7), where(6.9)
E
10
(
X
)
=
E
12
(
X
)
E
12
*
(
X
)
V
;
E
20
(
X
)
=
E
12
*
(
X
)
V
;
E
22
2
(
X
)
=
E
22
*
(
X
)
=
E
22
(
X
)
;
E
12
(
X
)
E
22
(
X
)
=
E
12
(
X
)
,and the operator V corresponds to
the representation(6.10)
J
|
ℋ
0
+
˙
ℋ
1
+
˙
ℋ
2
=
(
0
V
−
1
0
V
0
0
0
0
I
2
)
.Let 𝒳
0 be a family of
all sets under Condition (6.5). If X
,
Y
∈
𝒳
0, then, for the operators E
(
X
) and E
(
Y
), the condition of the form (6.8) holds
simultaneously. Since E
(
X
)
E
(
Y
)
=
E
(
Y
)
E
(
X
)
=
E
(
X
∩
Y
), then(6.11)
E
22
(
X
)
E
22
(
Y
)
=
E
22
(
Y
)
E
22
(
X
)
=
E
22
(
X
∩
Y
)
,
E
12
(
X
)
E
22
(
Y
)
=
E
12
(
Y
)
E
22
(
X
)
=
E
12
(
X
∩
Y
)
E
22
(
X
∩
Y
)
.Now, we discuss the spectrum multiplicity of the family E
˜
λ. Recall that a subspace 𝔏 is said to be cyclic with respect to E
˜
λ if CLin
λ
∈
[
−
1
;
1
]
∖
{
0
}
{
E
λ
𝔏
}
=
ℋ
˜.Definition 6.1.
In what follows, anonsingular multiplicity of J-orth.sp.f. E
λ means the
minimal dimension of all cyclic subspaces with respect to E
˜
λ.
Remark 6.2.
Due to
the choice ofℋ
˜ one can assume
that a cyclic subspace 𝔏 is taken such
that 𝔏
⊂
ℋ
2, so the nonsingular multiplicity of E
λ coincides with
the multiplicity (in the ordinary sense) of the (orthogonal) spectral function P
2
E
λ
|
ℋ
2.
Proposition 6.3.
If (6.1) and
(6.2) are fulfilled and the nonsingular multiplicity of E
λ is greater than dim
ℋ
1, then there is a decomposition ℋ
˜
=
ℋ
(
1
)
+
˙
ℋ
(
2
) such that ℋ
(
1
)
[
⊥
]
ℋ
(
2
), E
λ
ℋ
(
1
)
⊂
ℋ
(
1
), E
λ
ℋ
(
2
)
⊂
ℋ
(
2
) for every λ
∈
[
−
1
;
1
]
∖
{
0
}, the subspace ℋ
(
1
) is uniformly
positive and the J-orth.sp.f. E
λ
|
(
ℋ
(
1
)
)
[
⊥
] has nonsingular
multiplicity less than or equal to dim
ℋ
1.Proof.
Sinceℋ
1
⊂
ℋ
˜ and (6.2)
holds, there is a sequence of disjoint subsets {
X
k
}
1
∞ such that a
condition of the type (6.5) holds for all k
=
1
,
2
,
…, ∪
k
=
1
∞
X
k
=
[
−
1
;
1
]
∖
{
0
}, and for every k, the equality E
12
(
X
k
)
ℋ
2
=
ℋ
1 is true. It is
clear that dim
(
E
12
*
(
X
k
)
ℋ
1
)
=
dim
ℋ
1. Let {
u
j
(
k
)
}
1
n be some
orthonormalized basis in E
12
*
(
X
k
)
ℋ
1, k
=
1
,
2
,
…. Set(6.12)
u
j
=
∑
k
=
1
∞
(
1
/
2
)
k
u
j
(
k
)
,
j
=
1
,
2
,
…
,
n
;
ℋ
(
3
)
=
CLin
Y
∈
𝒳
0
{
E
22
(
Y
)
u
j
}
j
=
1
n
,
ℋ
(
1
)
=
(
ℋ
(
3
)
)
⊥
∩
ℋ
2, ℋ
(
2
)
=
CLin
Y
∈
𝒳
0
{
E
(
Y
)
ℋ
(
3
)
}. We show that the subspaces ℋ
(
1
) and ℋ
(
2
) are as desired.
First, we prove that, for every set X
∈
𝒳
0, the equality(6.13)
E
12
(
X
)
ℋ
(
1
)
=
{
0
}
.holds. Indeed (see (6.11)), if y
∈
ℋ
(
1
) and z
∈
ℋ
1, then (6.14)
(
E
12
(
X
)
y
,
z
)
=
(
y
,
E
12
*
(
X
)
z
)
=
(
y
,
∑
k
=
1
∞
E
22
(
X
k
)
E
21
*
(
X
)
z
)
=
∑
k
=
1
∞
(
y
,
E
22
(
X
∩
X
k
)
E
21
*
(
X
k
)
z
)
=
∑
k
=
1
∞
(
y
,
E
22
(
X
∩
X
k
)
∑
j
=
1
n
α
j
(
k
)
u
j
(
k
)
)
=
0.
Next, it is clear that ℋ
(
3
) is an invariant
subspace with respect to the operators E
22
(
X
), therefore, the subspace ℋ
(
1
) has the same
property. Thus, taking into account (6.13), one can conclude that ℋ
(
1
) is invariant
with respect to E
λ.Remark 6.4.
Proposition6.3 shows that, in some
problems one can assume that the nonsingular multiplicity of E
λ is finite. At
the same time, this hypothesis is not convenient in many cases because the
decomposition ℋ
˜
=
ℋ
(
1
)
+
˙
ℋ
(
2
) is not uniquely
defined and, moreover, the subspace ℋ
(
1
) can be always
extended saving all properties enumerated in Proposition 6.3. Basically,
Proposition 6.3 gives a possibility to illustrate peculiarities of J-orth.sp.f. in
question using J-orth.sp.f.
with finite nonsingular multiplicity.In addition
to (6.3), set (6.15)
ℋ
˜
↑
=
ℋ
0
⊕
ℋ
2
,
E
˜
λ
=
E
λ
|
ℋ
˜
,
E
˜
λ
↑
=
(
P
0
+
P
2
)
E
λ
|
ℋ
˜
↑
.Now, let L
σ
→
2
(
ℰ
) and M
σ
→
2
(
ℰ
) be some
standard Hilbert space and some standard space of measurable vector-functions,
respectively, and let g
˜
1
(
t
)
,
g
˜
2
(
t
)
,
…
,
g
˜
k
(
t
) be a finite collection
of vector functions such that(6.16)
(
a
)
g
˜
j
(
t
)
∈
M
σ
→
(
ℰ
)
,
j
=
1
,
2
,
…
,
k
;
(
b
)
the set
{
g
˜
j
(
t
)
}
j
=
1
k
is a set of unbounded elements from
L
σ
→
2
(
ℰ
)
linearly independent modulo
L
σ
→
2
(
ℰ
)
and conformed with
X
τ
.Below L
˜
σ
→
2
(
ℰ
)
⊂
M
σ
→
(
ℰ
) is the linear
span formed by L
σ
→
2
(
ℰ
) and the
collection {
g
˜
j
(
t
)
}
j
=
1
k, where (see Section 5) the functions from {
g
˜
j
(
t
)
}
j
=
1
k are assumed to
be normalized, pairwise orthogonal, and orthogonal to L
σ
→
2
(
ℰ
), that is, on L
˜
σ
→
2
(
ℰ
), a structure of Hilbert space is defined. Note that,
sometimes, L
˜
σ
→
2
(
ℰ
) will be
considered without its Hilbert structure as a vector subspace from M
σ
→
(
ℰ
) but this case
will be noted explicitly.


## 6.2. Functional Model ofE
λ (A Special Case)

Theorem 6.5.
If, forJ-orth.sp.f. E
λ, (6.1) and (6.2) hold, and for a canonical scalar
product (
⋅
,
⋅
), (6.4) holds, then there exist a standard Hilbert
space L
σ
→
2
(
ℰ
), a collection {
g
˜
j
(
t
)
}
j
=
1
k satisfying
(6.16) which generates the expansion L
˜
σ
→
2
(
ℰ
) of L
σ
→
2
(
ℰ
), and an isometric operator W: L
˜
σ
→
2
(
ℰ
)
↦
ℋ
˜, W
L
σ
→
2
(
ℰ
)
=
ℋ
2 such that, for
every λ
∈
[
−
1
;
1
]
∖
{
0
}, the following representations take place (see
(6.10)):(6.17)
E
˜
λ
=
W
⋅
X
λ
*
⋅
(
W
)
−
1
,
W
↑
=
(
I
2
⊕
V
)
W
,
E
˜
λ
↑
=
W
↑
⋅
X
λ
⋅
(
W
↑
)
−
1
,where X
λ
=
{
X
λ
,
L
˜
σ
→
2
(
ℰ
)
}, k
=
dim
ℋ
0
=
dim
ℋ
1
.
Proof.
First,P
2
E
λ
|
ℋ
2 is an
orthogonal (in the ordinary Hilbert sense) spectral function, so there is a
space L
σ
→
2
(
ℰ
) such that the
operator P
2
E
λ is similar to
the operator X
λ
=
{
X
λ
,
L
σ
→
2
(
ℰ
)
}. Let a space L
σ
→
2
(
ℰ
) be already
chosen and let an isometric operator W
2: L
σ
→
2
(
ℰ
)
↦
ℋ
2 be such that P
2
E
λ
|
ℋ
2
=
W
2
X
λ
W
2
−
1. Now, it is necessary to find a collection of
unbounded elements {
g
˜
j
(
t
)
}
j
=
1
k, so that the corresponding space L
˜
σ
→
2
(
ℰ
) and the
operator W would be as
desired. Choose, in ℋ
1, some orthonormalized basis {
e
j
}
j
=
1
k. Let Δ
⊂
[
−
1
;
1
]
∖
{
0
} be a closed
interval. Consider the expression (
E
(
Δ
)
x
,
e
j
), where x
∈
ℋ
2. It can be considered as a continuous linear
functional acting in ℋ
2 or,
equivalently, in L
σ
→
2
(
ℰ
). By the well-known theorem of Riesz on the general
representation of continuous linear functional, there is a function g
j
,
Δ
(
t
)
∈
L
σ
→
2
(
ℰ
) such that(6.18)
(
E
(
Δ
)
x
,
e
j
)
=
∫
−
1
1
(
f
(
t
)
,
g
j
,
Δ
(
t
)
)
ℰ
d
σ
(
t
)
,where f
(
t
)
=
W
2
−
1
x or (thanks to
properties of the spectral function)(6.19)
(
E
(
Δ
)
x
,
e
j
)
=
∫
Δ
(
f
(
t
)
,
g
j
,
Δ
(
t
)
)
ℰ
d
σ
(
t
)
,
(6.20)
g
j
,
Δ
(
t
)
=
0
,
if
t
∉
Δ
.Condition (6.20) means that there is a function g
˜
j
(
t
)
∈
M
σ
→
(
ℰ
) such that, for
every interval Δ as defined
above,(6.21)
g
j
,
Δ
(
t
)
=
χ
Δ
(
t
)
g
˜
j
(
t
)
,where χ
Δ
(
t
) is the
indicator of the interval Δ. The collection {
g
˜
j
(
t
)
}
j
=
1
k is as desired.
We show this. First, we prove that the set {
g
˜
j
(
t
)
}
j
=
1
k is linearly
independent modulo L
σ
→
2
(
ℰ
). Suppose the contrary, that is, suppose the existence
of a collection of coefficients {
γ
j
}
j
=
1
k such
that(6.22)
0
<
∑
j
=
1
k
|
γ
j
|
,
∑
j
=
1
k
γ
j
g
˜
j
(
t
)
∈
L
σ
→
2
(
ℰ
)
.Let y
=
∑
j
=
1
k
γ
j
e
j. By the definition of the space ℋ
˜, there is a sequence {
x
m
}
1
∞
⊂
Lin
{
E
(
Δ
)
ℋ
} such that ∥
x
m
−
y
∥
→
0 for m
→
∞. Set W
2
−
1
(
P
2
x
m
)
=
f
m
(
t
). Note that f
m
(
t
)
≡
0 in some
neighborhood of zero, therefore, the representation(6.23)
∥
x
m
−
y
∥
2
=
(
∥
f
m
(
t
)
∥
L
σ
→
2
(
ℰ
)
)
2
+
∑
j
=
1
k
|
γ
j
−
∫
−
1
1
(
f
m
(
t
)
,
g
˜
j
(
t
)
)
ℰ
d
σ
(
t
)
|
2is valid, hence lim
⁡
m
→
∞
∥
f
m
(
t
)
∥
=
0,(6.24)
lim
⁡
m
→
∞
∫
−
1
1
(
f
m
(
t
)
,
g
˜
j
(
t
)
)
ℰ
d
σ
(
t
)
=
γ
j
,
j
=
1
,
2
,
…
,
k
.These equalities and (6.22) imply(6.25)
∑
j
=
1
k
|
γ
j
|
2
=
lim
⁡
m
→
∞
∑
j
=
1
k
γ
¯
j
∫
−
1
1
(
f
m
(
t
)
,
g
˜
j
(
t
)
)
ℰ
d
σ
(
t
)
=
lim
⁡
m
→
∞
∫
−
1
1
(
f
m
(
t
)
,
∑
j
=
1
k
γ
j
g
˜
j
(
t
)
)
ℰ
d
σ
(
t
)
=
0
,
which is a contradiction, that is, (6.22) is impossible.
Second, formL
˜
σ
→
2
(
ℰ
) according to
the above procedure using the space L
σ
→
2
(
ℰ
) and the
collection of unbounded elements {
g
˜
j
(
t
)
}
j
=
1
k. Next, set(6.26)
W
f
(
t
)
=
W
2
f
(
t
)
,
if
f
(
t
)
∈
L
σ
→
2
(
ℰ
)
;
W
g
˜
j
(
t
)
=
e
j
,
j
=
1
,
2
,
…
,
k
.In this case the dual operator W
↑ is defined by
the equalities(6.27)
W
↑
f
(
t
)
=
W
2
f
(
t
)
,
if
f
(
t
)
∈
L
σ
→
2
(
ℰ
)
;
W
↑
g
˜
j
(
t
)
=
h
j
,
where
h
j
=
V
−
1
e
j
,
j
=
1
,
2
,
…
,
k
.Now, pass to (6.17). Assume (as above) that for a closed
interval Δ the condition Δ
⊂
[
−
1
;
1
]
∖
{
0
} holds. Then, in
concordance with (6.8) and (6.19), one has E
12
(
Δ
)
x
=
∑
j
=
1
k
(
x
,
z
j
(
Δ
)
)
e
j, where z
j
(
Δ
)
=
W
g
j
,
Δ
(
t
), j
=
1
,
2
,
…
,
k, so E
20
(
Δ
)
h
j
=
z
j
(
Δ
), j
=
1
,
2
,
…
,
k. The last equalities, (3.21), and (3.23) imply (see
the notation introduced in (6.15)) W
−
1
E
˜
↑
(
Δ
)
W
g
˜
j
(
t
)
=
χ
Δ
(
t
)
g
˜
j
(
t
), j
=
1
,
2
,
…
,
k. Due to the choice of L
σ
→
2
(
ℰ
), the rest follows from (6.9) and (6.27).
Definition 6.6.
IfJ-orth.sp.f. E
λ satisfies (6.1)
and (6.2), and the space L
˜
σ
→
2
(
ℰ
) has (6.17),
then L
˜
σ
→
2
(
ℰ
) is said to be a basic model space for E
λ (compatible
with (6.3) and (6.15)), and the operator W is said to be
an operator of similarity (generated by L
˜
σ
→
2
(
ℰ
)).Theorem 6.7.
Let a spaceL
˜
σ
→
2
(
ℰ
) be the
expansion of a standard Hilbert space L
σ
→
2
(
ℰ
) generated by a
collection {
g
˜
j
(
t
)
}
j
=
1
k satisfying
(6.16). Then there exist a Pontryagin space ℋ and a J-orth.sp.f. E
λ on ℋ such that L
˜
σ
→
2
(
ℰ
) is a basic
model space for E
λ.
Proof.
Letℋ be the set of
triples (
u
,
v
,
f
(
t
)
) with u
,
v
∈
ℂ
k and f
(
t
)
∈
L
σ
→
2
(
ℰ
). For x
1
=
(
u
1
,
v
1
,
f
1
(
t
)
) and x
2
=
(
u
2
,
v
2
,
f
2
(
t
)
), put(6.28)
(
x
1
,
x
2
)
=
(
u
1
,
u
2
)
ℂ
k
+
(
v
1
,
v
2
)
ℂ
k
+
(
f
1
(
t
)
,
f
2
(
t
)
)
L
σ
→
2
(
ℰ
)
,
[
x
1
,
x
2
]
=
(
u
1
,
v
2
)
ℂ
k
+
(
v
1
,
u
2
)
ℂ
k
+
(
f
1
(
t
)
,
f
2
(
t
)
)
L
σ
→
2
(
ℰ
)
.
Next, let {
e
j
}
j
=
1
k be the
canonical basis in ℂ
k. Then, for intervals Δ
=
[
α
,
β
), where 0
∉
Δ and α
≠
0, we put(6.29)
E
(
Δ
)
(
e
j
,
0,
0
)
=
(
0,
∑
m
=
1
k
(
χ
Δ
(
t
)
g
˜
j
(
t
)
,
χ
Δ
(
t
)
g
˜
m
(
t
)
)
L
σ
→
2
(
ℰ
)
⋅
e
m
,
χ
Δ
(
t
)
g
˜
j
(
t
)
)
for
j
=
1
,
2
,
…
,
k
,
E
(
Δ
)
(
0,
v
,
0
)
=
(
0,
0,
0
)
for every
v
∈
ℂ
k
,
E
(
Δ
)
(
0,
0,
f
(
t
)
)
=
(
0,
∑
m
=
1
k
(
χ
Δ
(
t
)
f
(
t
)
,
χ
Δ
(
t
)
g
˜
m
(
t
)
)
L
σ
→
2
(
ℰ
)
⋅
e
m
,
χ
Δ
(
t
)
f
(
t
)
)
.
Direct verification shows that E
(
Δ
) generates a J-orth.sp.f.
with (6.1) and (6.2). Let us check that(6.30)
ℋ
1
=
{
(
0
,
v
,
0
)
}
v
∈
ℂ
k
,
ℋ
2
=
{
(
0
,
0,
f
(
t
)
)
}
f
(
t
)
∈
L
σ
→
2
(
ℰ
)
.We denote, by the symbol (
L
σ
→
2
(
ℰ
)
)
0, the set of all functions from L
σ
→
2
(
ℰ
) vanishing near
zero, and for every f
(
t
)
∈
(
L
σ
→
2
(
ℰ
)
)
0, we set T
1
f
:
=
∑
m
=
1
k
(
f
(
t
)
,
g
˜
m
(
t
)
)
L
σ
→
2
(
ℰ
)
⋅
e
m. Then the space ℋ
˜ is the closure
of the set(6.31)
{
(
0,
T
1
f
,
f
(
t
)
)
}
f
(
t
)
∈
(
L
σ
→
2
(
ℰ
)
)
0
.Let a system {
v
j
}
j
=
1
k be an
orthonormalized basis in ℂ
k. Then(6.32)
v
j
=
∑
m
=
1
k
α
m
j
e
mand (6.31) can be rewritten as(6.33)
{
(
0,
∑
m
=
1
k
(
f
(
t
)
,
g
^
m
(
t
)
)
L
σ
→
2
(
ℰ
)
⋅
v
m
,
f
(
t
)
)
}
f
(
t
)
∈
(
L
σ
→
2
(
ℰ
)
)
0
,where, by (6.32), g
^
m
(
t
)
=
∑
j
=
1
k
α
¯
j
m
g
˜
j
(
t
)
. Note that
system {
g
^
m
(
t
)
}
m
=
1
k also has
(6.16).
Initially, the
basis{
v
m
}
m
=
1
k was arbitrary
but now we pass to choose it in a special way. Since the elements from {
g
˜
m
(
t
)
}
m
=
1
k are unbounded,
there is a sequence {
f
l
(
t
)
}
l
=
1
∞
⊂
(
L
σ
→
2
(
ℰ
)
)
0 such that lim
⁡
l
→
∞
∥
f
l
(
t
)
∥
L
σ
→
2
(
ℰ
)
=
0 and ∥
T
1
f
l
∥
ℂ
k
=
1 for every l. Let w
l
=
T
1
f
l. Since w
l
∈
ℂ
k, one can assume that there exists lim
⁡
l
→
∞
w
l, and put v
1
:
=
lim
⁡
l
→
∞
w
l
∈
ℋ
˜. Now, we apply the same scheme for defining v
2. Let T
2
f
:
=
∑
m
=
2
k
(
f
(
t
)
,
g
^
m
(
t
)
)
L
σ
→
2
(
ℰ
)
⋅
v
m
=
T
1
f
−
v
1
⋅
(
T
1
f
,
v
1
)
ℂ
k. Then (
0,
T
2
f
,
f
(
t
)
)
∈
ℋ
˜ for every f
(
t
)
∈
(
L
σ
→
2
(
ℰ
)
)
0. Since the elements from {
g
^
m
(
t
)
}
m
=
2
k are unbounded
for every choice of {
v
j
}
j
=
1
k, there exists a (new) sequence {
f
l
(
t
)
}
l
=
1
∞
⊂
(
L
σ
→
2
(
ℰ
)
)
0 such that lim
⁡
l
→
∞
∥
f
l
(
t
)
∥
L
σ
→
2
(
ℰ
)
=
0 and ∥
T
2
f
l
∥
ℂ
k
=
1 for every l. Let w
l
=
T
2
f
l. Since w
l
∈
ℂ
k, one can assume that there exists lim
⁡
l
→
∞
w
l, and put v
2
:
=
lim
⁡
l
→
∞
w
l
∈
ℋ
˜. Then v
1
⊥
v
2. The rest of the proof of (6.30) is now evident.
Next, we put W
f
(
t
)
=
(
0
,
0,
f
(
t
)
) for every f
(
t
)
∈
𝒥
−
L
σ
→
2
(
ℰ
), and W
g
˜
j
(
t
)
=
(
0
,
e
j
,
0
) for j
=
1
,
2
,
…
,
k. The rest is straightforward.We note that a basic model space for a given J-orth.sp.f. is
not uniquely defined and discusses the arbitrariness for the choice of a
concrete model. The construction of the basic model space was started from
(6.3) and these depend on the choice of a canonical scalar product. So the
first question is the following: does the basic model space depend on a choice
of canonical scalar product? In order to answer this question, note, first,
that ℋ
1 does not depend
on any canonical scalar product and is calculated using only the Hermitian
sesquilinear form [
⋅
,
⋅
] and J-orth.sp.f. E
λ. Second, one can consider the factor space ℋ
˜/ℋ
1. Then, by virtue of (6.2), the sesquilinear form [
⋅
,
⋅
] generates in it
the structure of Hilbert spaces, and the spectral function induced on ℋ
˜/ℋ
1 by the operator
function E
˜
λ is orthogonal.
Third, by Definition 4.2 and Proposition 4.3, this induced function is similar
to the spectral function P
2
E
λ
|
ℋ
2. Summarizing all mentioned arguments, we obtain the
following proposition.Proposition 6.8.
IfL
˜
σ
→
2
(
ℰ
) is a basic
model space for J-orth.sp.f. E
λ satisfying
(6.1) and (6.2), and W is the
corresponding operator of similarity, then the orth.sp.f. E
λ
∗, induced on the Hilbert factor space ℋ
∗
:
=
ℋ
˜
/
ℋ
1 by the J-orth.sp.f. E
λ, is similar to the operator family X
λ acting in L
σ
→
2
(
ℰ
), and the corresponding operator of similarity W
∗
:
L
σ
→
2
(
ℰ
)
↦
ℋ
∗ is induced by
the operator W.Thus the choice of L
σ
→
2
(
ℰ
) as a source for
constructing a basic model space L
˜
σ
→
2
(
ℰ
) is realized in
the same class of spaces as for the orth.sp.f. E
λ
∗ and does not
depend on a canonical scalar product on ℋ
˜.Proposition 6.9.
Let spacesL
˜
σ
→
2
(
ℰ
) and L
⌣
σ
→
2
(
ℰ
) be expansions
of a space L
σ
→
2
(
ℰ
) generating,
respectively, by collections {
g
˜
j
(
t
)
}
j
=
1
k and {
g
⌣
j
(
t
)
}
j
=
1
k with (6.16),
let both L
˜
σ
→
2
(
ℰ
) and L
⌣
σ
→
2
(
ℰ
) be a basic
model space for the same J-orth.sp.f. E
λ with (6.1) and
(6.2), and let W
(
1
) and W
(
2
) be the
corresponding operators of similarity. If for every function f
(
t
)
∈
L
σ
→
2
(
ℰ
), the relation(6.34)
(
W
(
1
)
−
W
(
2
)
)
f
(
t
)
∈
ℋ
1
,holds, then the spaces L
˜
σ
→
2
(
ℰ
) and L
⌣
σ
→
2
(
ℰ
) coincide as
vector subspaces in M
σ
→
(
ℰ
).
Proof.
The
complete proof will be divided in several stages. First, consider the
case(6.35)
W
(
1
)
L
σ
→
2
(
ℰ
)
=
W
(
2
)
L
σ
→
2
(
ℰ
)
.This equality means that, in particular, the basic model spaces L
˜
σ
→
2
(
ℰ
) and L
⌣
σ
→
2
(
ℰ
) were
constructed according to the procedure mentioned during the proof of Theorem
6.5 using the same decomposition ℋ
˜
=
ℋ
1
+
˙
ℋ
2, but, generally speaking, a different choice of
scalar products (
⋅
,
⋅
)
(
1
), (
⋅
,
⋅
)
(
2
), and (or) basis in ℋ
1. Put(6.36)
e
j
(
1
)
:
=
W
(
1
)
g
˜
j
(
t
)
,
e
j
(
2
)
:
=
W
(
2
)
g
⌣
j
(
t
)
,
j
=
1
,
2
,
…
,
k
.Since the Hilbert structure of the spaces L
˜
σ
→
2
(
ℰ
) and L
⌣
σ
→
2
(
ℰ
) is not in
question, only the choice of the bases {
e
j
(
1
)
}
1
k and {
e
j
(
2
)
}
1
k in ℋ
1 is important
(if scalar products are different then corresponding bases are necessarily
different). Let(6.37)
a
j
(
1
)
(
Δ
)
=
(
E
(
Δ
)
x
,
e
j
(
1
)
)
(
1
)
,
a
j
(
2
)
(
Δ
)
=
(
E
(
Δ
)
x
,
e
j
(
2
)
)
(
2
)
,
Δ
⊂
[
−
1
;
1
]
∖
{
0
}
,
x
∈
ℋ
2
,
e
j
(
2
)
=
∑
l
=
1
k
γ
j
l
e
l
(
1
)
,
j
=
1
,
2
,
…
,
k
.Then, by (6.19) and (6.21)(6.38)
a
j
(
1
)
(
Δ
)
=
∫
Δ
(
f
(
t
)
,
g
˜
j
(
t
)
)
ℰ
d
σ
(
t
)
,
a
j
(
2
)
(
Δ
)
=
∫
Δ
(
f
(
t
)
,
g
⌣
j
(
t
)
)
ℰ
d
σ
(
t
)
,where f
(
t
)
=
(
W
(
1
)
)
−
1
x
=
(
W
(
2
)
)
−
1
x. But {
a
j
(
1
)
(
Δ
)
}
1
k and {
a
j
(
2
)
(
Δ
)
}
1
k are the
coefficients of decomposition for the same vector with respect to bases {
e
j
(
1
)
}
1
k and {
e
j
(
2
)
}
1
k, respectively. Thus (6.37) gives the collection of
equalities a
l
(
2
)
(
Δ
)
=
∑
j
=
1
k
γ
j
l
a
j
(
1
)
(
Δ
), j
=
1
,
2
,
…
,
k, hence(6.39)
∫
Δ
(
f
(
t
)
,
g
⌣
l
(
t
)
)
ℰ
d
σ
(
t
)
=
∫
Δ
(
f
(
t
)
,
∑
j
=
1
k
γ
j
l
g
˜
j
(
t
)
)
ℰ
d
σ
(
t
)
.Since the segment Δ is arbitrary
and f
(
t
) runs through
the set of all functions from L
σ
→
2
(
ℰ
), this equality gives(6.40)
g
⌣
l
(
t
)
=
∑
j
=
1
k
γ
j
l
g
˜
j
(
t
)
,
j
=
1
,
2
,
…
,
k
.Thus
L
˜
σ
→
2
(
ℰ
)
⊃
L
⌣
σ
→
2
(
ℰ
). Since the spaces L
˜
σ
→
2
(
ℰ
) and L
⌣
σ
→
2
(
ℰ
) have an equal
status, the inverse inclusion is also true. So L
˜
σ
→
2
(
ℰ
) and L
⌣
σ
→
2
(
ℰ
) coincide as
subsets of M
σ
→
(
ℰ
). The proposition is proved under (6.35).
For the second
stage assume that the model spacesL
˜
σ
→
2
(
ℰ
) and L
⌣
σ
→
2
(
ℰ
) are constructed
using different decompositions of the space ℋ
˜, that is, ℋ
˜
=
ℋ
1
+
˙
ℋ
2
(
1
), ℋ
˜
=
ℋ
1
+
˙
ℋ
2
(
2
), but the basis in ℋ
1 and, moreover,
the scalar product in ℋ
1 for both spaces
are the same. Thus(6.41)
e
j
=
W
(
1
)
g
˜
j
(
t
)
=
W
(
2
)
g
⌣
j
(
t
)
,
j
=
1
,
2
,
…
,
k
.Next, the space ℋ
2
(
2
) can be
represented in the form(6.42)
ℋ
2
(
2
)
=
{
x
+
Q
x
}
x
∈
ℋ
2
(
1
)
,where Q
:
ℋ
2
(
2
)
↦
ℋ
1 is a linear
operator. It is clear that Q
x
=
∑
j
=
1
k
e
j
(
x
,
z
j
), where {
z
j
}
1
k is a vector set
from ℋ
2
(
1
). Note that the equality(6.43)
(
W
(
1
)
)
−
1
x
=
(
W
(
2
)
)
−
1
(
x
+
Q
x
)holds for all x
∈
ℋ
2
(
1
). Indeed, if f
(
t
)
:
=
(
W
(
1
)
)
−
1
x, then by (6.42), W
(
2
)
f
(
t
)
=
y
+
Q
y, where y
∈
ℋ
2
(
1
). This implies (
W
(
1
)
−
W
(
2
)
)
f
(
t
)
=
x
−
y
−
Q
y, and thanks to (6.34), one has x
=
y. Since W
(
2
) is a one-to-one
mapping, (6.43) is proved.
Now, letx
∈
ℋ
2
(
1
)
∩
(
E
(
Δ
)
ℋ
+
˙
ℋ
1
). Then by (6.15), the representation ℋ
˜
=
ℋ
1
+
˙
ℋ
2
(
1
) implies E
(
Δ
)
x
=
x
+
∑
j
=
1
k
e
j
(
x
,
s
j
(
1
)
), where (cf., (6.19))(6.44)
s
j
(
1
)
=
W
(
1
)
χ
Δ
(
t
)
g
˜
j
(
t
)
,
j
=
1
,
2
,
…
,
k
,so if (W
(
1
)
)
−
1
x
=
f
(
t
), then(6.45)
E
(
Δ
)
x
=
x
+
∑
j
=
1
k
e
j
∫
Δ
(
f
(
t
)
,
g
˜
j
(
t
)
)
ℰ
d
σ
(
t
)
.The last equality can be rewritten as (6.46)
E
(
Δ
)
x
=
(
x
+
Q
x
)
+
∑
j
=
1
k
e
j
(
x
,
s
j
(
1
)
−
z
j
)
=
(
x
+
Q
x
)
+
∑
j
=
1
k
e
j
∫
Δ
(
f
(
t
)
,
g
˜
j
(
t
)
−
ζ
j
(
t
)
)
ℰ
d
σ
(
t
)
,where ζ
j
(
t
)
=
(
W
(
1
)
)
−
1
z
j
∈
L
σ
→
2
(
ℰ
), j
=
1
,
2
,
…
,
k. On the other
hand, by (6.15) and (6.43), the representation ℋ
˜
=
ℋ
1
+
˙
ℋ
2
(
2
) and the
equality E
(
Δ
)
x
=
E
(
Δ
)
(
x
+
Q
x
) imply(6.47)
E
(
Δ
)
x
=
(
x
+
Q
x
)
+
∑
j
=
1
k
e
j
∫
Δ
(
f
(
t
)
,
g
⌣
l
(
t
)
)
ℰ
d
σ
(
t
)
.The comparison between (6.46) and (6.47) gives(6.48)
∑
j
=
1
k
e
j
∫
Δ
(
f
(
t
)
,
g
⌣
l
(
t
)
)
ℰ
d
σ
(
t
)
=
∑
j
=
1
k
e
j
∫
Δ
(
f
(
t
)
,
g
˜
j
(
t
)
−
ζ
j
(
t
)
)
ℰ
d
σ
(
t
)
.Since the function f
(
t
) is subjected
only to the condition f
(
t
)
=
χ
Δ
(
t
)
f
(
t
)
∈
L
σ
→
2
(
ℰ
) and the closed
segment Δ is arbitrary,
then g
⌣
j
(
t
)
=
g
˜
j
(
t
)
−
ζ
j
(
t
), j
=
1
,
2
,
…
,
k. The proposition is proved under Hypothesis (6.41).
The general case can be reduced to the two particular cases considered above,
so the rest is straightforward.The following proposition
is a partial inverse of Proposition6.9.Proposition 6.10.
Let spacesL
˜
σ
→
2
(
ℰ
) and L
⌣
σ
→
2
(
ℰ
) be the
expansions of a space L
σ
→
2
(
ℰ
) generated,
respectively, by collections {
g
˜
j
(
t
)
}
j
=
1
k and {
g
⌣
j
(
t
)
}
j
=
1
k satisfying
(6.16). If L
˜
σ
→
2
(
ℰ
) and L
⌣
σ
→
2
(
ℰ
) coincide as
vector subspaces in M
σ
→
(
ℰ
) and the space L
˜
σ
→
2
(
ℰ
) is a basic
model space for some J-orth.sp.f. E
λ with (6.1) and
(6.2), then the space L
⌣
σ
→
2
(
ℰ
) is also a basic
model space for E
λ.Proof.
LetW be the operator
of similarity for E
λ that
corresponds to the basic model space L
˜
σ
→
2
(
ℰ
). The scheme of the proof will be the same as for
Proposition 6.9; two particular cases will be considered and the superposition
of these will give the general case.
So let a system{
g
⌣
j
(
t
)
}
j
=
1
k be such
that(6.49)
g
˜
j
(
t
)
=
∑
l
=
1
k
γ
j
l
g
⌣
j
(
t
)
,
j
=
1
,
2
,
…
,
k
.Set ℋ
2
:
=
W
L
σ
→
2
(
ℰ
). The corresponding decomposition ℋ
˜
=
ℋ
1
+
˙
ℋ
2 (generated by W) will be taken
also for the construction of an operator of similarity W
′ corresponding
to L
⌣
σ
→
2
(
ℰ
). First, set W
′
f
(
t
)
:
=
W
f
(
t
) for f
(
t
)
∈
L
σ
→
2
(
ℰ
).
Next, forx
∈
ℋ
2, x
=
W
f
(
t
), one has(6.50)
P
1
E
(
Δ
)
x
=
∑
j
=
1
k
e
j
⋅
∫
Δ
(
f
(
t
)
,
g
˜
j
(
t
)
)
ℰ
d
σ
(
t
)
,this, jointly with (6.49) implies(6.51)
P
1
E
(
Δ
)
x
=
∑
j
=
1
k
e
j
′
⋅
∫
Δ
(
f
(
t
)
,
g
⌣
j
(
t
)
)
ℰ
d
σ
(
t
)
,where e
j
′
=
∑
l
=
1
k
γ
¯
l
j
e
l, j
=
1
,
2
,
…
,
k. The vector system {
e
j
′
}
j
=
1
k is taken as the
new orthonormalized basis in ℋ
1 (so, generally
speaking, the Hilbert structure on ℋ
1 is redefined).
Set W
′
g
⌣
j
(
t
)
:
=
e
j
′, j
=
1
,
2
,
…
,
k. Case (6.49) has been finished.
Now letg
˜
j
(
t
)
=
g
⌣
j
(
t
)
+
ζ
j
(
t
), ζ
j
(
t
)
∈
L
σ
→
2
(
ℰ
), j
=
1
,
2
,
…
,
k. Put(6.52)
W
′
f
(
t
)
=
W
f
(
t
)
+
∑
j
=
1
k
e
j
⋅
∫
−
1
1
(
f
(
t
)
,
ζ
j
(
t
)
)
ℰ
d
σ
(
t
)
,where f
(
t
)
∈
L
σ
→
2
(
ℰ
) and W
′
g
⌣
j
(
t
)
=
e
j, j
=
1
,
2
,
…
,
k
, that is, ℋ
2
′
=
W
′
L
σ
→
2
(
ℰ
). Note also that the Hilbert structure of ℋ
1 does not
change. It is easy to check that the operator W
′ and the
decomposition ℋ
˜
=
ℋ
1
+
˙
ℋ
2
′ are as
required.
Remark 6.11.
The proof of Proposition6.10 shows
that, if L
˜
σ
→
2
(
ℰ
) is a basic
model space for J-orth.sp.f. E
λ with (6.1) and
(6.2), W is a
corresponding operator of similarity, and L
˜
σ
→
2
(
ℰ
) and W are conformed
with the decomposition ℋ
˜
=
ℋ
1
+
˙
ℋ
2, where ℋ
2
=
W
L
˜
σ
→
2
(
ℰ
), then for every different decomposition ℋ
˜
=
ℋ
1
+
˙
ℋ
2
′ it is possible
to find a new Hilbert structure of the vector set L
˜
σ
→
2
(
ℰ
) (evidently this
change does not touch L
σ
→
2
(
ℰ
)) and a new
isometric map (an operator of similarity) W
′: L
˜
σ
→
2
(
ℰ
)
↦
ℋ
˜ such that W
′
L
σ
→
2
(
ℰ
)
=
ℋ
2
′ and (e.g.,
(6.34)) (W
−
W
′
)
L
σ
→
2
(
ℰ
)
⊂
ℋ
1. An analogous proposition can be formulated for a
change of the given Hilbert structure on ℋ
1.
Remark 6.12.
Propositions6.9 and 6.10 show that the Hilbert structure of the basic model
space L
˜
σ
→
2
(
ℰ
) introduced
during an expansion of L
σ
→
2
(
ℰ
) is not, in some
sense, really important. On the other hand, an example mentioned below
demonstrates that spaces L
˜
σ
→
2
(
ℰ
) and L
⌣
σ
→
2
(
ℰ
) obtained as
different expansions of a common space L
σ
→
2
(
ℰ
) can be basic
model spaces for the same J-orth.sp.f. E
λ and be
different vector subspaces in M
σ
→
2
(
ℰ
). Thus (6.34) in Proposition 6.9 cannot be omitted.
Example 6.13.
Letℋ be the totality
of triples (
γ
h
,
ξ
e
,
f
(
t
)
), where h and e are some fixed
abstract vectors γ
,
ξ
∈
ℂ, f
(
t
)
∈
L
t
2
(
ℂ
), so the measure defining the space L
t
2
(
ℂ
) is the standard
Lebesgue measure. For x
1
=
(
γ
1
h
,
ξ
1
e
,
f
1
(
t
)
) and x
2
=
(
γ
2
h
,
ξ
2
e
,
f
2
(
t
)
), put(6.53)
(
x
1
,
x
2
)
=
γ
1
γ
¯
2
+
ξ
1
ξ
¯
2
+
∫
−
1
1
f
1
(
t
)
f
¯
2
(
t
)
d
t
,
[
x
1
,
x
2
]
=
γ
1
ξ
¯
2
+
ξ
1
γ
¯
2
+
∫
−
1
1
f
1
(
t
)
f
¯
2
(
t
)
d
t
.
Next, let(6.54)
E
τ
x
=
(
0,
(
γ
∫
−
1
τ
(
1
/
t
2
)
d
t
+
∫
−
1
τ
(
1
/
t
)
d
t
)
e
,
χ
[
−
1
;
τ
)
(
t
)
(
(
γ
/
t
)
+
f
(
t
)
)
)
,for τ
∈
[
−
1
;
0
), and let (6.55)
E
τ
x
=
(
γ
h
,
(
ξ
−
∫
τ
1
(
γ
+
t
f
(
t
)
)
t
2
d
t
)
e
,
χ
[
−
1
;
τ
)
(
t
)
f
(
t
)
+
γ
(
χ
[
−
1
;
τ
)
(
t
)
−
1
)
t
)
,for τ
∈
(
0
;
1
].Direct verification
shows that E
τ is a J-orth.sp.f.
with (6.1), (6.2). In the capacity of a basic model space for E
τ, one can take, first, the space L
˜
t
2
(
ℂ
) generated by L
t
2
(
ℂ
) and g
˜
(
t
)
=
1
/
t, or, second, the space L
⌣
t
2
(
ℂ
) generated by L
t
2
(
ℂ
) and g
⌣
(
t
)
=
(
1
/
t
)
e
i
/
t. In the first case, W
(
ξ
g
˜
(
t
)
+
f
(
t
)
)
=
(
0
,
ξ
e
,
f
(
t
)
) and in the second
case, W
′
(
ξ
g
⌣
(
t
)
+
f
(
t
)
)
=
(
0
,
ξ
e
,
e
i
/
t
f
(
t
)
) (W and W
′ are the
corresponding operators of similarity). At the same time, the spaces L
˜
t
2
(
ℂ
) and L
⌣
t
2
(
ℂ
) form different
vector subsets in M
t
(
ℂ
) because |
lim
⁡
α
→
+
0
∫
α
1
g
˜
(
t
)
e
−
i
/
t
d
t
|
<
∞ and lim
⁡
α
→
+
0
∫
α
1
g
⌣
(
t
)
e
−
i
/
t
d
t
=
∞, that is, unbounded elements g
˜
(
t
) and g
⌣
(
t
) are linearly
independent modulo L
t
2
(
ℂ
).Theorem 6.14.
Let spacesL
˜
σ
→
2
(
ℰ
) and L
⌣
σ
→
2
(
ℰ
) be the
expansions of a space L
σ
→
2
(
ℰ
) generated,
respectively, by collections {
g
˜
j
(
t
)
}
j
=
1
k and {
g
⌣
j
(
t
)
}
j
=
1
k with (6.16).
Then L
˜
σ
→
2
(
ℰ
) and L
⌣
σ
→
2
(
ℰ
) are basic model
spaces for a common J-orth.sp.f. E
λ with (6.1) and
(6.2) if and only if there is a unitary-valued function U
t of the type
(3.23) such that the sets L
˜
σ
→
2
(
ℰ
) and {
U
t
f
⌣
(
t
)
}
f
⌣
(
t
)
∈
L
⌣
σ
→
2
(
ℰ
) coincide as
vector spaces in M
σ
→
(
ℰ
).
Proof.
Necessity By virtue of Remark 6.11, one can assume that, for model spaces L
˜
σ
→
2
(
ℰ
), L
σ
→
2
(
ℰ
), and corresponding operators of similarity W
(
1
) and W
(
2
), there exist the same representation ℋ
˜
=
ℋ
1
⊕
ℋ
2 and the same
Hilbert structure on ℋ
1. In this case, the operator 𝒰
:
L
˜
σ
→
2
(
ℰ
)
↦
L
˜
σ
→
2
(
ℰ
), 𝒰
:
=
(
W
(
2
)
)
−
1
W
(
1
) is well
defined. Direct verification gives 𝒰
{
X
τ
,
L
˜
σ
→
2
(
ℰ
)
}
*
=
{
X
τ
,
L
⌣
σ
→
2
(
ℰ
)
}
*
𝒰, which implies 𝒰
{
X
τ
,
L
˜
σ
→
2
(
ℰ
)
}
=
{
X
τ
,
L
⌣
σ
→
2
(
ℰ
)
}
𝒰. Since the subspace L
σ
→
2
(
ℰ
) is invariant
both for X
τ and 𝒰, these two operators commute on L
σ
→
2
(
ℰ
). Moreover, for every closed interval Δ
⊂
[
−
1
;
1
]
∖
{
0
}, the condition X
(
Δ
)
g
˜
j
(
t
)
∈
L
σ
→
2
(
ℰ
) holds for all g
˜
j
(
t
), j
=
1
,
2
,
…
k. The rest follows from Proposition 3.15.
Sufficiency
By
Theorem6.7, one can construct a Pontryagin space ℋ and a J-orth.sp.f. E
λ such that L
˜
σ
→
2
(
ℰ
) is a basic
model space for E
λ. The rest follows from (3.23) and Proposition 6.10.The results obtained above show that the choice of a
basic model spaceL
˜
σ
→
2
(
ℰ
) for J-orth.sp.f. E
λ with (6.1) and
(6.2) is reduced (up to a finite number of functions) to the choice of a
standard Hilbert space L
σ
→
2
(
ℰ
). The last choice is not uniquely defined. The
arbitrariness of the choice of L
σ
→
2
(
ℰ
) can be
diminished if one takes in consideration only the standard Hilbert spaces
orderly decomposable on uniform components. In the latter case, the choice is
reduced (see [25, Theorem VII.6]) to the
choice of scalar measure in the corresponding class of equivalent measures.
Note that, if L
σ
→
2
(
ℰ
) is orderly
decomposable on uniform components, then it is naturally embedded to the space M
σ
→
(
ℰ
) and, therefore,
this space is also orderly decomposable on uniform components. Moreover, the
space M
σ
→
(
ℰ
) does not depend
on σ
(
t
) but on the
corresponding class of equivalent measures. This reasoning leads to the
following theorem.Theorem 6.15.
LetL
˜
σ
→
2
(
ℰ
) be a basic
model space for J-orth.sp.f. E
λ with (6.1) and
(6.2), and let the space L
σ
→
2
(
ℰ
) be orderly
decomposable on uniform components. Then every basic model space for E
λ constructed as
an expansion of a standard Hilbert space orderly decomposable on uniform
components has the following form;(6.56)
{
1
φ
(
t
)
U
t
f
˜
(
t
)
}
f
˜
(
t
)
∈
L
˜
σ
→
2
(
ℰ
)
,where φ
(
t
) is a fixed
scalar function satisfying the conditions φ
(
t
)
≥ 0 a.e. on [
−
1
;
1
], and ∫
−
1
1
φ
2
(
t
)
d
σ
(
t
)
<
∞, and the operator-valued function U
t is subject to
(3.24). Conversely, every functional vector space of the form (6.56) is a basic
model space for E
λ.
Remark 6.16.
Let us
consider an interpretation of Theorem6.15 for the case of J-orth.sp.f. E
λ with simple
nonsingular multiplicity (in particular, E
λ can be cyclic
although this condition is not necessary). In this case, L
σ
→
2
(
ℰ
) converts to the
space L
σ
2
(
ℂ
) of scalar
functions, and U
t converts to the
multiplication operator by a function e
i
ϑ
(
t
), where ϑ
(
t
)
:
[
−
1
;
1
]
↦
[
0
;
2
π
) is a μ
σ-measurable
function. Next, let {
g
˜
j
(
t
)
}
j
=
1
k be a collection
of unbounded elements generating the expansion L
˜
σ
2
(
ℂ
) that is a basic
model space for E
λ. Then every basic model space for E
λ constructed as
an expansion of a standard Hilbert space orderly decomposable on uniform
components has, as a subset of M
σ
(
ℂ
), the following form:(6.57)
{
1
φ
(
t
)
(
f
(
t
)
+
e
i
ϑ
(
t
)
∑
j
=
1
k
γ
j
g
˜
j
(
t
)
)
}
f
(
t
)
∈
L
σ
2
(
ℂ
)
,
γ
1
,
…
,
γ
k
∈
ℂ
.Concluding this part, let us consider an estimation of the norm of a J-orthogonal
projection E
(
X
), where the set X satisfies (6.5)
and the conventions given above on the choice of the canonical scalar product
on ℋ are
preserved.Proposition 6.17.
If, under the mentioned above
conditions,E
(
X
)
≠
0, then ∥
E
(
X
)
∥
=
1
+
∥
E
12
(
X
)
∥
2.
Proof.
IfE
12
(
X
)
=
0, when the desired formula directly follows from
(6.8), (6.9), so we assume that E
12
(
X
)
≠
0. Let(6.58)
V
−
1
x
∈
ℋ
0
,
y
∈
ℋ
2
,
∥
x
∥
2
+
∥
y
∥
2
=
1.Then ∥
E
(
X
)
∥
2
=
sup
⁡
x
,
y
{
∥
E
(
X
)
(
V
−
1
x
+
y
)
∥
2
}. On the other hand taking into account (6.9), (6.11),
we have ∥
E
(
X
)
(
V
−
1
x
+
y
)
∥
2
=
∥
E
12
(
X
)
E
12
*
(
X
)
x
+
E
12
(
X
)
y
∥
2
+
∥
E
12
*
(
X
)
x
+
E
22
(
X
)
y
∥
2
≤
(
1
+
∥
E
12
(
X
)
∥
2
)
⋅
∥
E
12
*
(
X
)
x
+
E
12
(
X
)
y
∥
2. A first step is an estimation of ∥
E
12
*
(
X
)
x
+
E
22
(
X
)
y
∥
2. Since E
22
(
X
) is an
orthogonal projection, it is clear that it is enough to consider the case E
22
(
X
)
y
=
y. Next, ∥
E
12
*
(
X
)
x
+
y
∥
≤
∥
E
12
*
(
X
)
∥
∥
x
∥
+
∥
y
∥
≤
∥
E
*
(
X
)
∥
2
+
1
⋅
∥
x
∥
2
+
∥
y
∥
2
=
∥
E
*
(
X
)
∥
2
+
1. Thus(6.59)
∥
E
(
X
)
(
V
−
1
x
+
y
)
∥
2
≤
(
1
+
∥
E
12
(
X
)
∥
2
)
2
.The proof will be finished if vectors x and y, satisfying (6.58) and converting (6.59) to an
equality, can be found.
Since the
operatorE
12
*
(
X
) is finite
dimensional, there is a vector z
≠
0, z
∈
ℋ
1 such
that(6.60)
∥
E
12
*
(
X
)
z
∥
=
∥
E
12
*
(
X
)
∥
∥
z
∥
.Let(6.61)
x
=
∥
E
12
(
X
)
∥
z
∥
z
∥
∥
E
*
(
X
)
∥
2
+
1
,
y
=
E
12
*
(
X
)
z
∥
E
12
*
(
X
)
z
∥
∥
E
*
(
X
)
∥
2
+
1
.Since E
22
(
X
)
E
12
*
(
X
)
=
E
12
*
(
X
) (see (6.9), y)is well
defined. Then(6.62)
E
12
*
(
X
)
x
+
y
=
(
∥
E
12
(
X
)
∥
+
1
∥
E
12
*
(
X
)
∥
)
⋅
E
12
(
X
)
z
∥
z
∥
∥
E
*
(
X
)
∥
2
+
1
=
∥
E
*
(
X
)
∥
2
+
1
⋅
E
12
(
X
)
z
∥
z
∥
⋅
∥
E
12
*
(
X
)
∥
2
.
Finally (see (6.60)), ∥
E
12
(
X
)
E
12
*
(
X
)
z
∥
=
∥
E
12
(
X
)
∥
∥
E
12
*
(
X
)
z
∥.Proposition 6.18.
LetL
˜
σ
→
2
(
ℰ
) be a basic
model space for J-orth.sp.f. E
λ with (6.1) and
(6.2), and let {
g
˜
j
(
t
)
}
j
=
1
k be the
corresponding system with (6.16). Then(6.63)
1
k
∫
X
∑
j
=
1
k
∥
g
˜
j
(
t
)
∥
ℰ
2
d
σ
(
t
)
≤
∥
E
12
(
X
)
∥
2
≤
∫
X
∑
j
=
1
k
∥
g
˜
j
(
t
)
∥
ℰ
2
d
σ
(
t
)
.
Proof.
Let{
e
j
}
j
=
1
k be the
orthonormalized basis in ℋ
1 corresponding
to the system {
g
˜
j
(
t
)
}
j
=
1
k. Then the operator E
12
*
(
X
)
E
12
(
X
) has, with
respect to this basis the matrix representation (γ
m
j
)
k
×
k with the
elements γ
m
j
=
∫
X
(
g
˜
m
(
t
)
,
g
˜
j
(
t
)
)
ℰ
d
σ
(
t
). The rest is a usual estimation for the norm of a
positive matrix.


## 6.3. A Functional Model forE
λ
(The General Case)

In the previous
subsection, we studiedJ-orth.sp.f. E
λ with (6.1),
(6.2). Now, we turn to a more general case dropping (6.2). It implies that
(6.9) and (6.11) must be substituted by some new ones. Let us conserve (6.3),
(6.15). However, it is necessary to take into account that, now, generally
speaking, the subspace ℋ
2 is indefinite.
Recall that J-orth.sp.f. E
λ belongs to the
class D
κ
+, so there is an E
λ-invariant pair
of J-orthogonal
maximal semi-definite pseudoregular subspaces 𝔏
+ and 𝔏
− with
finite-dimensional isotropic part; moreover, due to (3.15)(b), we can assume
that, for every closed interval Δ
⊂
[
−
1
;
1
]
∖
{
0
}, the subspace (
E
(
Δ
)
ℋ
)
∩
𝔏
+ is positive and
the subspace (
E
(
Δ
)
ℋ
)
∩
𝔏
− is negative.
Thanks to the hypothesis, the following subspaces are well
defined:(6.64)
ℋ
˜
+
=
CLin
Δ
⊂
[
−
1
;
1
]
∖
{
0
}
{
E
(
Δ
)
𝔏
+
}
,
ℋ
˜
−
=
CLin
Δ
⊂
[
−
1
;
1
]
∖
{
0
}
{
E
(
Δ
)
𝔏
−
}
.Set(6.65)
ℋ
2
+
=
ℋ
2
∩
ℋ
˜
+
,
ℋ
2
−
=
ℋ
2
∩
ℋ
˜
−
,and assume that a fundamental scalar product on ℋ is
simultaneously canonical for the subspace ℋ
˜
⊕
ℋ
0 and, on the
last space, compatible (see Definition 3.4) with the given decomposition of the
subspaces(6.66)
ℋ
1
+
˙
ℋ
2
+
,
ℋ
1
+
˙
ℋ
2
−
.Thus(6.67)
J
|
ℋ
0
+
˙
ℋ
1
+
˙
ℋ
2
=
(
0
V
−
1
0
V
0
0
0
0
J
2
)
,where the operator V is the same as
in (6.10), and J
2 is a canonical
symmetry of the form [
⋅
,
⋅
] on ℋ
2. Then, for X satisfying
(6.5), the elements of the matrix realization(6.68)
E
(
X
)
|
ℋ
0
+
˙
ℋ
1
+
˙
ℋ
2
=
(
0
0
0
E
10
(
X
)
0
E
12
(
X
)
E
20
(
X
)
0
E
22
(
X
)
)satisfy (cf. (6.9)) the following relations(6.69)
E
10
(
X
)
=
E
12
(
X
)
J
2
E
12
*
(
X
)
V
;
E
20
(
X
)
=
J
2
*
E
12
(
X
)
V
;
E
22
2
(
X
)
=
E
22
*
(
X
)
=
E
22
#
(
X
)
=
E
22
(
X
)
;
E
12
(
X
)
E
22
(
X
)
=
E
12
(
X
)
.Here, E
22
#
(
X
) is the operator J-adjoint to the
operator E
22
(
X
) (note that now ℋ
2 is a Krein
space). We note also that (6.11) remains valid.Now, let𝒥
−
L
σ
→
2
(
ℰ
) be a standard
Krein space (see Section 3) and let {
g
˜
j
(
t
)
}
j
=
1
k be a system of
unbounded elements conformed with the operator-valued function X
τ and linearly
independent modulo 𝒥
−
L
σ
→
2
(
ℰ
) (i.e., {
g
˜
j
(
t
)
}
j
=
1
k satisfies
(6.16)). Denote, by 𝒥
−
L
˜
σ
→
2
(
ℰ
), the linear span generated by the space 𝒥
−
L
σ
→
2
(
ℰ
) and the system {
g
˜
j
(
t
)
}
j
=
1
k. Define, on 𝒥
−
L
˜
σ
→
2
(
ℰ
), structures of Hilbert and Krein spaces in the
following way: on 𝒥
−
L
σ
→
2
(
ℰ
) both structures
coincide with the original structures, and functions of the system {
g
˜
j
(
t
)
}
j
=
1
k are, by
definition, positive (as elements of the Krein space), mutually orthogonal and J-orthogonal,
normalized and J-normalized,
and orthogonal and J-orthogonal to 𝒥
−
L
σ
→
2
(
ℰ
). The space 𝒥
−
L
˜
σ
→
2
(
ℰ
) is said to be the expansion of
𝒥
−
L
σ
→
2
(
ℰ
) (generated
by the collection
{
g
˜
j
(
t
)
}
j
=
1
k).Theorem 6.19.
If aJ-orth.sp.f. E
λ satisfies (6.1)
(but not (6.2)) and a scalar product on ℋ is compatible
with (6.66), then there are, first, a subspace 𝒥
−
L
σ
→
2
(
ℰ
) and a system {
g
˜
j
(
t
)
}
j
=
1
k with (6.16)
forming together the space 𝒥
−
L
˜
σ
→
2
(
ℰ
), and, second, an isometric J-isometric
operator W: 𝒥
−
L
˜
σ
→
2
(
ℰ
)
↦
ℋ
˜, W
L
σ
→
2
(
ℰ
)
=
ℋ
2, such that, for every λ
∈
[
−
1
;
1
],(6.70)
E
˜
λ
=
W
⋅
X
λ
#
⋅
(
W
)
−
1
,
W
↑
=
(
I
2
⊕
V
)
W
,
E
˜
λ
↑
=
W
↑
⋅
X
λ
⋅
(
W
↑
)
−
1
,where X
λ
=
{
X
λ
,
𝒥
−
L
˜
σ
→
2
(
ℰ
)
}, k
=
dim
ℋ
0
=
dim
ℋ
1
.
Proof.
Since the scalar product(
⋅
,
⋅
) on ℋ
2 is canonical
and conformed with the decomposition ℋ
2
=
ℋ
2
+
[
+
˙
]
ℋ
2
−, the spectral function P
2
E
λ is
simultaneously orthogonal and J-orthogonal.
Then for P
2
E
λ, by Proposition 4.3 and in concordance with
Definition 4.2, there exist a model space 𝒥
−
L
σ
→
2
(
ℰ
) and an operator
of similarity W
2. The rest of the proof is analogous to the
corresponding stage of the proof for Theorem 6.5 taking into account that (6.9)
must be substituted by (6.69).Remark 6.20.
Under the conditions of Theorem6.19, the
operator function E
˜
λ is similar not
only to the operator function {
X
λ
#
,
𝒥
−
L
˜
σ
→
2
(
ℰ
)
} but also to the
operator-function {
X
λ
*
,
L
⌣
σ
→
2
(
ℰ
)
}, where the space L
⌣
σ
→
2
(
ℰ
) is formed by
the standard Hilbert space L
σ
→
2
(
ℰ
) (it is the same
as for 𝒥-L
˜
σ
→
2
(
ℰ
)) and the
system of unbounded elements {
g
⌣
j
(
t
)
=
𝒥
g
˜
j
(
t
)
}
j
=
1
k. Thus the properties of the operator-valued function E
˜
λ (as well as for E
˜
λ
↑
) do not depend on
either (6.2) is fulfilled or not for the operator-valued function E
λ. However this dependence exists for the J-orth.sp.f. E
λ, and this fact is connected with the different
structure of the operator E
10
(
X
) in (6.9) and
(6.69). Note, in particular, that, if (6.2) is not fulfilled, then the
estimation for the norm of operator E
12
(
X
) given in
Proposition 6.18 remains true, but at the same time, the equality for ∥
E
(
X
)
∥ from
Proposition 6.17 is, generally speaking, incorrect.
Example 6.21.
Take the spaceℋ coinciding with l
2, denote {
u
m
}
m
=
1
∞ the canonical
basis of this space, that is, u
1
=
(
1
,
0
,
0
,
…
), u
2
=
(
0
,
1
,
0
,
…
)
,
…, and define the canonical symmetry J by the
equalities J
u
0
=
u
1, J
u
1
=
u
0, J
u
2
m
+
1
=
u
2
m
+
1, J
u
2
m
=
−
u
2
m, m
=
1
,
2
,
…, and the spectral function E
λ by the
relations(i)
E
λ
=
0 for every λ
<
0;
(ii)
ifλ
>
0, then (
I
−
E
λ
)
u
0
=
∑
1
/
m
∈
[
λ
;
1
)
(
u
2
(
m
−
1
)
−
u
2
m
−
1
), (
I
−
E
λ
)
u
1
=
0, (
I
−
E
λ
)
u
2
(
m
−
1
)
=
u
2
(
m
−
1
)
+
u
1, (
I
−
E
λ
)
u
2
m
−
1
=
u
2
m
−
1
+
u
1 for 1
/
m
∈
[
λ
;
1
), and (
I
−
E
λ
)
u
2
(
m
−
1
)
=
(
I
−
E
λ
)
u
2
m
−
1
=
0 for 1
/
m
∉
[
λ
;
1
).For Δ
m
=
[
1
/
m
;
1
], one has ∥
E
12
(
Δ
m
)
∥
=
2
(
m
−
1
) and ∥
E
(
Δ
m
)
∥
=
2
m
−
1. Note that, here, the values ∥
E
12
(
Δ
m
)
∥ and ∥
E
(
Δ
m
)
∥ have the same
order of growth for m
→
∞. Note also that E
10
(
Δ
)
=
0 for every
closed segment Δ
⊂
[
−
1
;
1
]
∖
{
0
}.Definition 6.22.
If, for (6.3) and
(6.65), a relation between a J-orth.sp.f. E
λ satisfying
(6.1) and a space 𝒥
−
L
˜
σ
→
2
(
ℰ
) is given by
(6.70), then 𝒥
−
L
˜
σ
→
2
(
ℰ
) is said to be a basic model space for E
λ (compatible
with (6.3), (6.15), and (6.66)) and the operator W is said to be an operator of similarity corresponding to
this space.Note that Definition 6.22 does not
contradict Definition 6.6, but amplifies it; if, for J-orth.sp.f. E
λ, (6.2) is fulfilled, then 𝒥
=
I, so the canonical scalar product is uniquely defined
on ℋ
2 and coincides
with [
⋅
,
⋅
].Now, we show
that every space𝒥
−
L
˜
σ
→
2
(
ℰ
) can be
considered as a basic model space for a suitable J-orth.sp.f. E
λ
∈
D
k
+.Theorem 6.23.
Let a space𝒥
−
L
˜
σ
→
2
(
ℰ
) be the
expansion of a standard Krein space 𝒥
−
L
σ
→
2
(
ℰ
) generated by a
collection {
g
˜
j
(
t
)
}
j
=
1
k satisfying
(6.16). Then there exist a Krein space ℋ and a J-orth.sp.f. E
λ on ℋ such that E
λ
∈
D
k
+ and 𝒥
−
L
˜
σ
→
2
(
ℰ
) is a basic
model space for E
λ.
Proof.
Letℋ be the set of
triples (
u
,
v
,
f
(
t
)
) with u
,
v
∈
ℂ
k and f
(
t
)
∈
𝒥
−
L
σ
→
2
(
ℰ
). For x
1
=
(
u
1
,
v
1
,
f
1
(
t
)
) and x
2
=
(
u
2
,
v
2
,
f
2
(
t
)
), put (see
(4.3))(6.71)
(
x
1
,
x
2
)
=
(
u
1
,
u
2
)
ℂ
k
+
(
v
1
,
v
2
)
ℂ
k
+
(
f
1
(
t
)
,
f
2
(
t
)
)
L
σ
→
2
(
ℰ
)
,
[
x
1
,
x
2
]
=
(
u
1
,
v
2
)
ℂ
k
+
(
v
1
,
u
2
)
ℂ
k
+
[
f
1
(
t
)
,
f
2
(
t
)
]
L
σ
→
2
(
ℰ
)
.
Next, let {
e
j
}
j
=
1
k be the
canonical basis in ℂ
k. For the intervals Δ
=
[
α
,
β
), where 0
∉
Δ and α
≠
0, we put(6.72)
E
(
Δ
)
(
e
j
,
0,
0
)
=
(
0,
∑
m
=
1
k
[
χ
Δ
(
t
)
g
˜
j
(
t
)
,
χ
Δ
(
t
)
g
˜
m
(
t
)
]
L
σ
→
2
(
ℰ
)
⋅
e
m
,
χ
Δ
(
t
)
g
˜
j
(
t
)
)
for
j
=
1
,
2
,
…
,
k
,
E
(
Δ
)
(
0,
v
,
0
)
=
(
0,
0,
0
)
for every
v
∈
ℂ
k
,
E
(
Δ
)
(
0,
0,
f
(
t
)
)
=
(
0,
∑
m
=
1
k
[
χ
Δ
(
t
)
f
(
t
)
,
χ
Δ
(
t
)
g
˜
m
(
t
)
]
L
σ
→
2
(
ℰ
)
⋅
e
m
,
χ
Δ
(
t
)
f
(
t
)
)
.
Direct verification shows that E
(
Δ
) generates a J-orth.sp.f.
satisfying (6.1). The rest of this proof looks like the corresponding part of
the proof of Theorem 6.7 with some evident modifications.Next, the basic model space is constructed not only via the J-orth.sp.f. E
λ itself and via
the subspaces ℋ
˜ and ℋ
1 naturally
generated by E
λ (note that, in
this case, it is convenient to consider the mentioned subspaces as linear
topological spaces), but also via some other subspaces and via the Hilbert
structure introduced on ℋ
˜ and ℋ
1 that can be
defined with some ambiguity.Proposition 6.24.
Let a space𝒥
−
L
˜
σ
→
2
(
ℰ
) (i.e., the expansion
of standard Krein space 𝒥
−
L
σ
→
2
(
ℰ
) generated by a
collection {
g
˜
j
(
t
)
}
j
=
1
k with (6.16)) be
a basic model space for a J-orth.sp.f. E
λ compatible with
(6.3), (6.15), and (6.66). Then for every different
decomposition (6.73)
ℋ
˜
=
ℋ
1
+
˙
ℋ
2
′
,
ℋ
2
′
=
𝔏
+
[
+
˙
]
𝔏
−
,where 𝔏
+ and 𝔏
− are,
respectively, uniformly positive and uniformly negative subspace, and ℋ
1
+
˙
𝔏
+ and ℋ
1
+
˙
𝔏
− are E
λ-invariant,
there is a collection {
g
⌣
j
(
t
)
}
j
=
1
k with (6.16)
such that the expansion 𝒥-L
⌣
σ
→
2
(
ℰ
) of 𝒥
−
L
σ
→
2
(
ℰ
) generated by {
g
⌣
j
(
t
)
}
j
=
1
k is also a basic
model space for E
λ compatible with
(6.3), (6.15), and (6.73).
Proof.
Note
that a collection of unbounded elements{
g
⌣
j
(
t
)
}
j
=
1
k is uniquely
defined by a set composed by the following components: a choice of a normalized
basis in ℋ
1, a decomposition ℋ
˜
=
ℋ
1
+
˙
ℋ
2
′, and an isometric J-isometric map W
2
′: 𝒥
−
L
σ
→
2
(
ℰ
)
↦
ℋ
2
′. This collection always exists for the mentioned set,
so it is enough to show the existence of W
2
′. Consider two cases.
(1) Letℋ
2
′
=
ℋ
2. Thus, on the Krein space ℋ
2, there are two different P
2
E
λ-invariant
canonical decompositions. Then, thanks to Proposition 3.8 and (6.70)(6.74)
P
2
E
λ
|
ℋ
2
=
D
{
ℋ
2
+
,
ℋ
2
−
}
,
{
𝔏
+
,
𝔏
−
}
W
2
X
λ
W
2
−
1
(
D
{
ℋ
2
+
,
ℋ
2
−
}
,
{
𝔏
+
,
𝔏
−
}
)
−
1
,so one can take W
2
′
=
D
{
ℋ
2
+
,
ℋ
2
−
}
,
{
𝔏
+
,
𝔏
−
}
W
2.
(2) Letℋ
2
′
≠
ℋ
2 and let the
pairs {
ℋ
2
+
,
ℋ
2
−
} and {
𝔏
+
,
𝔏
−
} be related by a
condition of the type (3.9). Then by Remark 3.6 and
Proposition 3.7, one can take W
2
′
=
ℳ
ℋ
˜
,
ℋ
2
,
ℋ
2
′
⋅
W
2.
Now, let us
pass to the general case. Let𝔏
+
′
:
=
ℳ
ℋ
˜
,
ℋ
2
,
ℋ
2
′
ℋ
2
+ and 𝔏
−
′
:
=
ℳ
ℋ
˜
,
ℋ
2
,
ℋ
2
′
ℋ
2
−. Then ℋ
2
′
=
𝔏
+
′
[
+
˙
]
𝔏
−
′, and one can take(6.75)
W
2
′
=
D
{
𝔏
+
′
,
𝔏
−
′
}
,
{
𝔏
+
,
𝔏
−
}
ℳ
ℋ
˜
,
ℋ
2
,
ℋ
2
′
⋅
W
2
.
Corollary 6.25.
Assume thatE
λ is a J-orth.sp.f.
satisfying (6.1), (6.3) is fixed, and 𝒥
−
L
σ
→
2
(
ℰ
) is a standard
Krein space. Then, independently of any realization of (6.73), an expansion of 𝒥
−
L
σ
→
2
(
ℰ
), that is a basic model space for E
λ compatible with
(6.73), exists if and only if there is a J-isometric
operator W
2
:
𝒥
−
L
σ
→
2
(
ℰ
)
↦
ℋ
2 such that(6.76)
P
2
E
λ
|
ℋ
2
=
W
2
X
λ
W
2
−
1
.
Remark 6.26 (cf., with Proposition6.8).
Letℋ
∗ be the
factor-space of ℋ generated by ℋ
1. Then ℋ
∗ is a Krein
space and E
˜
λ induces, on
this space, the J-orth.sp.f. E
λ
∗. Note that the unique peculiarity of E
λ
∗ is regular and
can be removed, so, as a slight abuse of terminology, one can say that E
λ
∗ has no
peculiarities. It is clear that W
2 in (6.76)
exists if and only if there exists a J-isometric
operator W
∗
:
𝒥
−
L
σ
→
2
(
ℰ
)
↦
ℋ
∗ such that E
λ
∗
=
W
∗
X
λ
W
∗
−
1. Taking this reasoning and Corollary 6.25 into
account, one can say that an arbitrariness in the choice of a basic model space
for J-orth.sp.f. E
λ satisfying
(6.1) does not depend on the choice of P
2 but essentially
depends on an arbitrariness in the choice of a basic model space for J-orth.sp.f. E
λ
∗ with the empty
set of peculiarities. The last problem was considered in Section 4.
Proposition 6.27.
Assume that spaces𝒥
−
L
˜
σ
→
2
(
ℰ
) and 𝒥
′
−
L
⌣
σ
→
2
(
ℰ
) are constructed
on the base of the same space L
σ
→
2
(
ℰ
) passing by the
following way: first, introduce on it two (maybe different) sesquilinear forms
by fundamental symmetries 𝒥 and 𝒥
′; second, expand these two standard Krein spaces using
collections of unbounded elements {
g
˜
j
(
t
)
}
j
=
1
k and {
g
⌣
j
(
t
)
}
j
=
1
k, respectively. Assume also that, first, 𝒥
−
L
˜
σ
→
2
(
ℰ
) and 𝒥
′
−
L
⌣
σ
→
2
(
ℰ
) are basic model
spaces for a J-orth.sp.f. E
λ, second, W
(
1
) and W
(
2
) are
corresponding operators of similarity, and, third, every function f
(
t
)
∈
L
σ
→
2
(
ℰ
) satisfies
(6.34). Then 𝒥
=
𝒥
′ and the spaces 𝒥
−
L
˜
σ
→
2
(
ℰ
), and 𝒥
′
−
L
⌣
σ
→
2
(
ℰ
) coincide as
subsets in M
σ
→
(
ℰ).
Proof.
Condition (6.34) implies that [
W
(
1
)
f
(
t
)
,
W
(
1
)
f
(
t
)
]
=
[
W
(
2
)
f
(
t
)
,
W
(
2
)
f
(
t
)
] is true for
every f
(
t
)
∈
L
σ
→
2
(
ℰ
), so [
f
(
t
)
,
f
(
t
)
]
𝒥
−
L
˜
σ
→
2
(
ℰ
)
=
[
f
(
t
)
,
f
(
t
)
]
𝒥
−
L
⌣
σ
→
2
(
ℰ
). Hence 𝒥
=
𝒥
′. The rest of the reasoning is similar to that of
Proposition 6.9.Comparing the last proposition
with Proposition 6.10, note that the fulfilment of (6.34) is essential for the simultaneous
coincidence 𝒥
−
L
˜
σ
→
2
(
ℰ
) with 𝒥
′
−
L
⌣
σ
→
2
(
ℰ
) as subsets in M
σ
→
(
ℰ
), and 𝒥
−
L
σ
→
2
(
ℰ
) with 𝒥
′
−
L
σ
→
2
(
ℰ
) as standard
Krein spaces, but at the same time is not necessary.Theorem 6.28.
Assume
that spaces𝒥
−
L
˜
σ
→
2
(
ℰ
) and 𝒥
′
−
L
⌣
σ
→
2
(
ℰ
) are constructed
on the base of the same space L
σ
→
2
(
ℰ
) passing by the
following way: first, introduce on it two (maybe different) sesquilinear forms
by fundamental symmetries 𝒥 and 𝒥
′, and second, expand these two standard Krein spaces
using collections of unbounded elements {
g
˜
j
(
t
)
}
j
=
1
k and {
g
⌣
j
(
t
)
}
j
=
1
k, respectively. Then 𝒥
−
L
˜
σ
→
2
(
ℰ
) and 𝒥
′
−
L
⌣
σ
→
2
(
ℰ
) are basic model
spaces for a common J-orth.sp.f. E
λ satisfying
(6.1) if and only if there exist operator functions U
t and M
t, t
∈
[
−
1
;
1
], with (3.24) and (4.7), respectively, such that the
sets L
˜
σ
→
2
(
ℰ
) and {
M
t
U
t
f
˜
(
t
)
}
f
(
t
)
∈
L
˜
σ
→
2
(
ℰ
) coincide as
subsets of M
σ
→
(
ℰ
).The proof of this
theorem is similar to that of Theorem 6.14.


## 6.4. The Passage toJ-Symmetric Operator Families

Up to this
point, we discussed model representations not for a family𝔜
∈
D
κ
+ but for a J-orth.sp.f. E
λ with the unique
spectral peculiarity in zero subjected to the condition(6.77)
There exists
ℒ
+
∈
𝔐
+
(
ℋ
)
∩
h
+
such that for every
λ
∈
ℝ
∖
{
0
}
it is true
E
λ
𝔏
+
⊂
𝔏
+
and for every closed interval
Δ
⊂
ℝ
∖
{
0
}
there is the decomposition
E
(
Δ
)
ℋ
=
E
(
Δ
)
𝔏
+
[
+
˙
]
E
(
Δ
)
𝔏
−
.Now, let 𝔜
∈
D
κ
+ be a J-symmetric
commutative operator family. Then its linear span contains a family (not
uniquely defined) 𝔜
r of J-s.a. operators
such that the linear span of 𝔜
r coincides with
the linear span of 𝔜. In [20], it was proved that there is the decomposition ℋ
=
ℋ
r
e
[
+
˙
]
ℋ
i
m, where ℋ
r
e and ℋ
i
m are 𝔜-invariant, all
operators from 𝔜
r
|
ℋ
r
e have real
spectra, dim
ℋ
i
m
<
∞, and, at least, one J-s.a. operator
from Alg
𝔜
|
ℋ
i
m has no real
points in its spectrum. Let E
λ be an e.s.f.
for 𝔜
r
|
ℋ
r
e. Extend the action of E
λ for all ℋ setting E
λ
|
ℋ
i
m
=
0. This extended function E
λ is said to be an e.s.f. of 𝔜. Comparing Theorem 3.11 and Theorem 6.19, it is easy
to obtain the following result.Theorem 6.29.
If𝔜
∈
D
κ
+ is a
commutative J-symmetric
family, its e.s.f. E
λ satisfies
(6.1), a canonical scalar product on ℋ is compatible
with (6.66), 𝒥
−
L
˜
σ
→
2
(
ℰ
) is a basic
model space for E
λ, and W is a
corresponding operator of similarity, then, for every operator A
∈
𝔜, there is a function φ
(
t
) such
that(6.78)
A
˜
↑
=
W
↑
⋅
Φ
⋅
(
W
↑
)
−
1
,
A
˜
=
W
⋅
Φ
¯
#
⋅
W
−
1
,where A
˜
↑
:
=
(
P
0
⊕
P
1
)
A
|
ℋ
˜
↑, the space ℋ
˜
↑ and the
operator W
↑ are defined via
(6.15) and (6.70), Φ
=
{
Φ
,
𝒥
−
L
σ
→
2
(
ℰ
)
}, and Φ
¯ is the
multiplication operator by φ
¯
(
t
) also acting in
the space 𝒥
−
L
˜
σ
→
2
(
ℰ
).
Remark 6.30.
If,
under the conditions of Theorem6.29, J-orth.sp.f. E
λ satisfies
(6.2), then the space 𝒥
−
L
˜
σ
→
2
(
ℰ
) must be
substituted by the space L
˜
σ
→
2
(
ℰ
), and, respectively, the operator Φ
¯
# must be
substituted by the operator Φ
¯
*.
Definition 6.31.
If𝔜
∈
D
κ
+ is a
commutative J-symmetric
family and E
λ (satisfying
(6.1)) is its e.s.f., then a basic model space for E
λ is said to be
also a basic model space for 𝔜.
Remark 6.32.
Assume
that𝔜
∈
D
κ
+ is a
commutative J-symmetric
family, its e.s.f. E
λ satisfies
(6.1), and 𝒥
−
L
˜
σ
→
2
(
ℰ
) and W are,
respectively, a basic model space and a corresponding operator of similarity
for 𝔜. If, under these conditions, an operator A
∈
𝔜 and a function φ
(
t
) are related by
(6.78), then, according to Definition 3.12, the function φ
(
t
) is a portrait
of the operator A (with respect
to E
λ
), and the operator A is an original
(not unique in the general case) of φ
(
t
) in 𝔜. It is clear that the portrait of an operator does
not depend on the choice of the basic model space 𝒥
−
L
˜
σ
→
2
(
ℰ
) and can be
found via (3.15)(c), but at the same time depends on the choice of E
λ.
Remark 6.33.
Theorem6.29 brings a natural problem concerning a characterization of functions that
can be portraits for operators from a given commutative J-symmetric
operator family 𝔜
∈
D
κ
+ with a fixed
choice of J-orth.sp.f. E
λ. A partial answer to the problem is contained by
Propositions 5.4 and 5.5. Indeed, under the present conditions, the
function ν
(
t
) from (5.5) has
the form(6.79)
ν
(
t
)
=
{
∫
−
1
t
G
(
t
)
d
σ
(
t
)
,
t
∈
[
−
1
;
0
)
,
−
∫
t
1
G
(
t
)
d
σ
(
t
)
,
t
∈
(
0
;
1
]
,
where
G
(
t
)
=
1
+
∑
j
=
1
k
∥
g
˜
j
(
t
)
∥
ℰ
2
.


## 7. Closing Remarks

Spectral
functions are a traditional source for constructions of model spaces for
different classes of normal operators in Hilbert spaces, so it is natural to
use the same approach in Krein spaces. The first theorem concerning the
existence of spectral functions forπ-s.a. operators
was published by Kreĭn and Langer in [28] (see also [6] for detailed proofs). The spectral functions
introduced in Section 3.2 are a particular case of generalized spectral
functions [1]. Our
definition is inspired by spectral measures arising in the operator theory in
indefinite metric spaces (see [3, 6, 8, 29]) but formally,
independent (see [30])
of this Theory. Another development related with generalized spectral functions
can be found in [31–34]. There are some works on
model representations for self-adjoint operators and algebras in Pontryagin
spaces (the majority of them consider the case with the rank of indefiniteness
1) [35–40], (see also [41] for more references). In
[12], the case of a
self-adjoint cyclic operator acting in an arbitrary Pontryagin space was
considered. The class D
κ
+ that is
investigated in the present paper differs from the well-known class of
definitizable operators (see [42] for discussion) and represents another natural
generalization of the class of self-adjoint operators in Pontryagin spaces. The
notion of unbounded elements was given by the author in [43], thereupon, this notion was
used in [9], where
the existence of a basic model space for a single π-self-adjoint
operator was proved. The latter result was applied in [10] (devoted to a description
of a broad functional calculus for π-self-adjoint
operators) and [11],
where the bicommutant problem for a π-self-adjoint
operator was studied. Some remarks concerning a model representation of π-unitary
operators were given in [44]. A connection between the spaces of the type L
σ
∞
∩
L
ν
2 and operator
algebras was pointed out for the first time in [45] (for Pontryagin spaces with
the rank of indefiniteness 1) and [46]. On this subject, see also [47, 48]. Although the concept of
basic model spaces for commutative families of π-self-adjoint
or J-self-adjoint
operators of the D
κ
+-class was used
by the author in some previous papers, its complete description and
corresponding proofs as well as the discussion of the ambiguity of basic model
spaces are given here for the first time.



---

*Source: 439781-2008-05-05.xml*