# On Certain Aspects of the MultidimensionalS-Spectral Theory

Authors: Cristina Serbanescu; Ioan Bacalu
Journal: Mathematical Problems in Engineering
 (2014)

---

## Abstract

This paper has two parts. The first one provides the preliminary notions introducing certain general concepts, in order to study, in the second part, the properties of some operator systems which admit spectral residual decompositions,S-decomposable, S-spectral, and A
S-scalar systems, and so forth. The results obtained by Frunză, 1975, are generalized, taking the results of Foias, 1963, as a model and adopting them.


---

## Body

## 1. Introduction

Across this paper we will try to generalize for operators systems some of the results obtained by Vasilescu in [1] for a single operator: residual single valued extension property, analytic residuum, the problem of local spectra, and so forth.Most of the proofs are adaptations of the ones from [2, 3] with minor changes.All operators with a reasonable spectral decomposition haveS-decomposable restrictions and quotients. Subnormal, subscalar, and subdecomposable operators being restrictions and quotients of normal, scalar, and decomposable operators are thus S-decomposable (in fact S-normal, S-scalar, etc.). Eschmeier and Putinar [4] have shown that hyponormal operators are subscalar and therefore S-decomposable. Operators that admit scalar dilations (extensions) (C. Ionescu-Tulcea) or A-scalar (E. Stroescu) are S-decomposable. In fact, Eschmeier and Putinar [4] have shown that any operator is the quotient or restriction of a quotient of a decomposable operator and thus is S-decomposable or similar to an S-decomposable operator.Colojoară and Foiaş in [5, Chapter 6, Proposition 5
b] formulated an open problem: any decomposable operator is strongly decomposable (meaning are that the restrictions and quotients in relation to the spectral maximal space decomposable?). A partial answer was given in [6]: the operators with the spectrum of dimension ≤1 thus situated on a curve are strongly decomposable, more specifically if the spectrum is of dimension ≤1 and any of its subsets, in the relative topology, included, has a border ≤0, sets of class C. There are subsets in the C plane which do not have this property. The result was strengthened by the example given by Albrecht and Vasilescu [7]. Some of the results obtained for one operator were generalized for systems of operators.LetX be a Banach space and let B
X be the algebra of all linear bounded operators on X.Furthermore, letS
X be the family of all closed linear subspaces of X, let S
⊂
C
n be a compact set, and let F
S be the family of all closed subsets F
⊂
C
n that have the property: either F
∩
S
=
Ø or F
⊃
S.AnS-spectral capacity is an application E
:
F
S
→
S
X verifying the following properties:(1)
E
Ø
=
0, E
C
n
=
X;
(2)
E
⋂
i
=
1
∞
F
i
=
⋂
i
=
1
∞
E
F
i, for any series F
i
i
∈
N
⊂
F
S;
(3)
for any open finite S-covering {
G
S
}
∪
{
G
j
}
j
=
1
m of C
n, we have X
=
E
(
G
-
S
)
+
∑
j
=
1
m
E
(
G
-
j
).A commuting system of operatorsa
=
a
1
,
a
2
,
…
,
a
n
⊂
B
X is said to be S-decomposable if there is an S-spectral capacity such that(4)
a
i
E
F
⊂
E
F, for any F
∈
F
S and any i
=
1,2
,
…
,
n;
(5)
σ
a
,
E
F
⊂
F, for any F
∈
F
S.In case thatS
=
Ø, the S-spectral capacity is said to be spectral capacity, and the system is decomposable [8].


## 2. Systems of Commuting Operators

This section provides the preliminary notions introducing certain general concepts necessary for the study of theS-decomposable systems in Section 3.Definition 1.
Leta
=
a
1
,
a
2
,
…
,
a
n
⊂
B
X be a commuting operators system and S
a
⊂
C
n a compact minimal set having the property that H
n
-
1
C
∞
G
,
X
,
α
⊕
∂
-
=
0 for any open G
⊂
C
n with G
∩
S
a
=
Ø (minimal means that S
a is the intersection of all compact sets having the specified property). One will denote by d
a
,
x the union of all open sets V
⊂
C
n with the property that there exists a form ψ
∈
Λ
n
-
1
σ
∪
d
z
-
,
C
∞
V
,
X satisfying the equality s
x
=
α
⊕
∂
-
ψ meaning that
(1)
x
s
1
∧
s
2
∧
⋯
∧
s
n
=
z
1
-
a
1
s
1
+
⋯
+
z
n
-
a
n
s
n
∂
∂
z
-
n
W
W
+
∂
∂
z
-
1
d
z
-
1
+
⋯
+
∂
∂
z
-
n
d
z
-
n
∧
ψ
z
(we recall that there exist sets V with this property, e.g., the solving set r
a
,
X). We will also denote
(2)
g
a
,
x
=
C
n
∖
d
a
,
x
,
r
a
,
x
=
d
a
,
x
∩
C
n
∖
S
a
,
sp
a
,
x
=
C
n
∖
r
a
,
x
=
g
a
,
x
∪
S
a
.
The set r
a
,
x will be said to be theresolvent set of x related to a, sp
a
,
x will be said to be thespectrum of x related to a, and S
a will be called theresidual spectrum of a.
We will callanalytic resolvent set of x related to a and we will denote by ρ
a
,
x the set ρ
a
,
x
=
δ
a
,
x
∩
C
n
∖
S
a, where δ
a
,
x is the set of z
∈
C
n for which there exists an open neighbourhood V of z and n analytic function on V taking values in X, f
1
,
f
2
,
…
,
f
n satisfying the identity
(3)
x
=
ζ
1
-
a
1
f
1
ζ
+
⋯
+
ζ
n
-
a
n
f
n
ζ
,
ζ
∈
V
.
We will understand through theanalytic spectrum of x related to a the set
(4)
σ
a
,
x
=
C
n
∖
ρ
a
,
x
=
γ
a
,
x
∪
S
a
,
where
(5)
γ
a
,
x
=
C
n
∖
δ
a
,
x
.
We will prove that for an operators system that admits a spectral S-capacity we have
(6)
g
a
,
x
=
γ
a
,
x
,
d
a
,
x
=
δ
a
,
x
,
ρ
a
,
x
=
r
a
,
x
,
sp
a
,
x
=
σ
a
,
x
.Proposition 2.
For a commuting operators systema
=
a
1
,
a
2
,
…
,
a
n
⊂
B
X one has the following:(1)
x
=
0 implies g
a
,
x
=
Ø, s
p
a
,
x
=
S
a;
(2)
g
a
,
x
+
y
⊂
g
a
,
x
∪
g
a
,
y, s
p
a
,
x
+
y
⊂
s
p
a
,
x
∪
s
p
a
,
y
∀
x, y
∈
X;
(3)
g
a
,
b
y
⊂
g
a
,
x, s
p
a
,
b
y
⊂
s
p
a
,
x if b
a
i
=
a
i
b, b
∈
B
X, x
∈
X;
(4)
g
a
,
y
⊂
s
p
a
,
y
⊂
σ
a
,
Y,where Y is a (linear, closed) subspace of X invariant to all a
i and y
∈
Y.Proof.
(1°) follows from the fact that, forx
=
0 and any neighbourhood V
⊂
C
n, the form ψ
=
0
∈
Λ
n
-
1
[
σ
∪
d
z
-
,
C
∞
(
V
,
X
)
] verifies the relation s
x
=
α
⊕
∂
-
ψ meaning that
(7)
x
s
1
∧
s
2
∧
⋯
∧
s
n
=
z
1
-
a
1
s
1
+
⋯
+
z
n
-
a
n
s
n
∂
∂
z
-
n
W
W
+
∂
∂
z
-
1
d
z
-
1
+
⋯
+
∂
∂
z
-
n
d
z
-
n
∧
ψ
z
.
Let z
∈
d
a
,
x
∩
d
a
,
y and z
∈
V
i such that there exist the forms
(8)
ψ
i
∈
Λ
n
-
1
σ
∪
d
z
-
,
C
∞
V
i
,
X
,
i
=
1,2
verifying the equalities s
x
=
α
⊕
∂
-
ψ
1, s
y
=
α
⊕
∂
-
ψ
2. Then the form
(9)
ψ
1
+
ψ
2
∈
Λ
n
-
1
σ
∪
d
z
-
,
C
∞
V
1
∪
V
2
,
X
verifies the equality
(10)
s
x
+
y
=
α
⊕
∂
-
ψ
1
+
ψ
2
,
and hence (2°) is verified. The inclusions from (3°) result from the fact that, by considering the form ψ
∈
Λ
n
-
1
σ
∪
d
z
-
,
C
∞
V
,
X such that s
x
=
α
⊕
∂
-
ψ and by applying operator b to the coefficients of ψ, its commuting with each a
i  (
i
=
1,2
,
…
,
n
) implies
(11)
α
⊕
∂
-
b
ψ
=
b
α
⊕
∂
-
ψ
=
b
x
s
(admitting the equality on components). The inclusion (4°) and the remark on the resolvent set r
a
,
Y lead to the equality x
s
=
α
⊕
∂
-
ψ.Lemma 3.
LetV
1, V
2 be two open sets in C
n such that V
1
∩
V
2
≠
Ø.
Then for anyf
∈
C
∞
(
V
1
∩
V
2
,
X
) there exist f
i
∈
C
∞
V
i
,
X
(
i
=
1,2
) such that f
=
f
1
-
f
2 on V
1
∩
V
2 ([8], 1.2.1.).Lemma 4.
Leta
=
a
1
,
a
2
,
…
,
a
n
⊂
B
X be an operators system with the residual spectrum S
a and V
i
(
i
=
1,2
) two open sets in C
n
∖
S
a such that there exist the forms ψ
i
∈
Λ
n
-
1
σ
∪
d
z
-
,
C
∞
V
i
,
X with the property that s
x
=
α
⊕
∂
-
ψ
i on V
i. Then there exists a form
(12)
ψ
∈
Λ
n
-
1
σ
∪
d
z
-
,
C
∞
V
1
∪
V
2
,
X
having the following property: s
x
=
α
⊕
∂
-
ψ on V
1
∪
V
2.Proof.
WhenV
1
∩
V
2
=
Ø we can consider ψ
z
=
ψ
i
z for z
∈
V
i
(
i
=
1,2
) and we have s
x
=
α
⊕
∂
-
ψ on V
1
∪
V
2. If V
1
∩
V
2
≠
Ø we have α
⊕
∂
-
ψ
2
-
ψ
1
=
0 on V
1
∩
V
2. Since V
1
∩
V
2
=
G
⊂
C
n
∖
S
a, it results that there exists a form φ
∈
Λ
n
-
2
σ
∪
d
z
-
,
C
∞
V
1
∩
V
2
,
X such that
(13)
ψ
2
-
ψ
1
=
α
⊕
∂
-
φ
.
Indeed, the nucleus of the cofrontier operator α
⊕
∂
- is:
(14)
Ker
⁡
α
⊕
∂
-
:
Λ
n
-
1
σ
∪
d
z
-
,
C
∞
V
1
∩
V
2
,
X
WW
Λ
n
-
1
⟶
Λ
n
σ
∪
d
z
-
,
C
∞
V
1
∩
V
2
,
X
W
=
Im
⁡
⁡
α
⊕
∂
-
:
Λ
n
-
2
σ
∪
d
z
-
,
C
∞
V
1
∩
V
2
,
X
WWWW
⟶
Λ
n
-
1
σ
∪
d
z
-
,
C
∞
V
1
∩
V
2
,
X
.
By applying the preceding lemma to the coefficients of φ there follows φ
=
φ
1
-
φ
2 where φ
i
∈
Λ
n
-
2
σ
∪
d
z
-
,
C
∞
V
i
,
X
(
i
=
1,2
).
Consequently,(15)
α
⊕
∂
-
φ
1
-
φ
2
=
α
⊕
∂
-
φ
=
φ
2
-
φ
1
,
where ψ
1
+
α
⊕
∂
-
ψ
1
=
ψ
2
+
α
⊕
∂
-
ψ
2 on V
1
∩
V
2. By putting ψ
i
′
=
ψ
i
+
α
⊕
∂
-
φ
i
(
i
=
1,2
) we will obtain s
x
=
α
⊕
∂
-
ψ
i
′ on V
i
(
i
=
1,2
) and ψ
1
′
=
ψ
2
′ on V
1
∩
V
2. Hence by defining ψ
z
=
ψ
i
′
z for z
∈
V
i
(
i
=
1,2
) one obtains a form as the one required in the text of the lemma. The lemma is proved.Corollary 5.
LetV
i
i
=
1
m be a finite family of open sets from C
n
∖
S
a such that the equation s
x
=
α
⊕
∂
-
ψ has a solution ψ on each of them. If K
⊂
r
a
,
x is a compact set, there exists an open neighbourhood V of K
(
V
⊂
r
a
,
x
) on which the equation s
x
=
α
⊕
∂
-
ψ has a solution.Proof.
LetK
ν
ν
=
1
∞ be a growing sequence of compact sets such that r
a
,
x
=
⋃
ν
=
1
∞
K
ν. We will prove that there exists a corresponding sequence of forms ψ
ν
∈
Λ
n
-
1
σ
∪
d
z
-
,
C
∞
r
a
,
x
,
X that verify the equality s
x
=
α
⊕
∂
-
ψ
ν on a neighbourhood of K
ν. Then ψ
=
lim
⁡
ν
→
∞
⁡
ψ
ν exists and it is a global solution. We will start with K
1 (see [9, 10]).
By Corollary5 there exists a form ψ
1 defined in an open neighbourhood of K
1 and satisfying the equality s
x
=
α
⊕
∂
-
ψ
1
* on this neighbourhood. Since the space C
∞
r
a
,
x
,
X is invariant to multiplication by scalar functions of a C
∞ class ([3], 2.16.1) we can assume, without limiting the generality, that ψ
1
* is defined on r
a
,
x; indeed, by multiplying ψ
1
* by a suitable scalar function, the new form can be extended to r
a
,
x and we will obtain a form ψ
1 on r
a
,
x verifying the equality s
x
s
=
α
⊕
∂
-
ψ
1 on a neighbourhood of K
1. We will now suppose that the forms ψ
1
,
ψ
2
,
…
,
ψ
i from the desired sequences were already determined and let us determine ψ
i
+
1.
According to the preceding corollary there exists a neighbourhoodV
i
+
1 of the set K
i
+
1 and a form ψ
i
+
1 defined on this neighbourhood satisfying the equality s
x
=
α
⊕
∂
-
ψ
i
+
1
*, and we are allowed to suppose moreover that ψ
i
+
1
* is defined on the whole r
a
,
x. But s
x
=
α
⊕
∂
-
ψ
i on a vicinity V
i of K
i, and hence by subtraction we obtain α
⊕
∂
-
ψ
i
+
1
*
-
ψ
i
=
0 on V
i
∩
V
i
+
1; since V
i
∩
V
i
+
1
⊂
C
n
∖
S
a, it will result that there exists a form φ
′ such that ψ
i
+
1
*
-
ψ
i
=
α
⊕
∂
-
φ
′ on V
i
∩
V
i
+
1, and we may suppose that φ
′ is defined on r
a
,
x. We will put ψ
i
+
1
=
ψ
i
+
1
*
-
α
⊕
∂
-
φ
′ and obtain a form defined on r
a
,
x equal to ψ
i on V
i
∩
V
i
+
1 and satisfying the equality s
x
=
α
⊕
∂
-
φ
i
+
1 on the neighbourhood V
i
+
1 of K
i
+
1. By this the demonstration ends.Remark 6.
A local version of the Cauchy-Weil formula ([8, 1.2.4]) can be established on the same way as in [8, formula 1.5.1].
Leta
=
a
1
,
a
2
,
…
,
a
n
⊂
B
X be a commuting operators system with the residual spectrum S
a and U an open neighbourhood of sp
a
,
x; obviously U
⊃
S
a.
We will prove that there exists a formχ
∈
Λ
n
[
σ
∪
d
z
-
,
C
0
∞
(
C
n
,
X
)
] in the same cohomology class related to α
⊕
∂
- as s
x such that support (
χ
)
∈
U.
It follows that there exists a formψ
∈
Λ
n
-
1
σ
∪
d
z
-
,
C
∞
r
a
,
x
,
X such that s
x
=
α
⊕
∂
-
ψ. Let U
1 and U
2 be two open neighbourhoods relatively compact of sp
a
,
x, such that sp
a
,
x
⊂
U
1
⊂
U
-
1
⊂
U
2
⊂
U
-
⊂
U, and let us consider scalar C
∞-function h on C
∞, h
=
1 outside U
2, and h
=
0 on U
1. By using h let us define the form ψ
~ by ψ
~
=
h
ψ on r
a
,
x and ψ
~
=
0 on U
1.
This form has the coefficients inC
∞
C
n
,
X and satisfies the condition s
x
=
α
⊕
∂
-
ψ
~ outside the relatively compact set U
2. Hence by setting χ
=
s
x
-
α
⊕
∂
-
ψ
~ we will obtain a form defined on C
n with support χ
⊂
U
-
2
⊂
U that is precisely the form having the specified properties. Considering formula [8, 1.2.4.] and using form χ above we can write
(16)
x
=
1
2
π
i
n
∫
U
-
1
n
π
χ
∧
d
z
1
∧
⋯
∧
d
z
n
which will yield the local version of Cauchy-Weil formula.


## 3. Some Properties ofS-Decomposable Systems

For the Banach spaceX and for an arbitrary open set U
⊂
C
n, we denote by U
U
,
X the space of all X-valued analytic functions on U.Proposition 7.
Leta
=
a
1
,
a
2
,
…
,
a
n
⊂
B
X be an S-decomposable system, let D be an open polydisk with D
∩
S
=
Ø, let p be an integer, 0
≤
p
≤
n
-
1, and let ψ
∈
Λ
p
σ
,
U
D
,
X such that σ
ψ
=
0 where α is defined by
(17)
α
ψ
z
=
z
1
-
a
1
s
1
+
z
2
-
a
2
s
2
W
W
+
⋯
+
z
n
-
a
n
s
n
∧
ψ
z
.
Then for any polydisk D
′
⊂
D with D
-
′
⊂
D there exists a form ψ
∈
Λ
p
-
1
σ
,
U
D
′
,
X such that ψ
=
α
φ on D
′.The proof of Proposition 2.1.3. presented in [8] is also true in this case, with a single comment that D is not anymore any polydisk of C
n, but a polydisk that does not cross S.Theorem 8.
Ifa
=
a
1
,
a
2
,
…
,
a
n
⊂
B
X is S-decomposable, then S
⊃
S
a.Proof.
With minor differences, the proof is identical with the one for the decomposable systems(
S
=
Ø
) ([8, Proposition 2.1.4.]) where S
a
=
Ø is called property L. It will have to show that for any polydisk U
⊂
C
n such that U
∩
S
=
Ø we have H
i
U
U
,
X
,
α
=
0
(
0
≤
i
≤
n
-
1
).
We note thatH
i
U
U
,
X
,
α
=
0 implies H
i
C
∞
G
,
X
,
α
⊕
∂
-
=
0
(
0
≤
i
≤
n
-
1
) where U is any open polydisk from C
n and G is any open set G
⊂
C
n such that U
∩
S
=
Ø, G
∩
S
=
Ø; the proof is given in [8, Theorem 1.5.16.] for any U, G
⊂
C
n.
One motivates this through induction oni, beginning with i
=
0. Let f
∈
U
U
,
X such that α
f
=
0; according to the preceding proposition we will have f
=
0 on any polydisk D
′ with D
′
⊂
U and f
=
0 on U. Suppose that for any open polydisk D
⊂
C
n with D
∩
S
=
Ø we have H
i
-
1
U
D
,
X
,
α
=
0 with i fixed, 0
≤
i
≤
n
-
1, and let us prove that H
i
U
U
,
X
,
α
=
0.
LetD
ν be a sequence of polydisks, D
ν
∩
S
=
Ø, such that D
ν
⊂
D
-
ν
+
1 for any ν with ⋃
ν
=
1
∞
D
ν
=
U and ψ
∈
Λ
i
σ
,
U
U
,
X such that α
ψ
=
0. By applying the preceding proposition for D
2, we infer that there exists a form φ
1
∈
Λ
i
-
1
σ
,
U
D
2
,
X such that ψ
=
α
φ
1 on D
2; analogously we can find a form φ
2
′ on D
3 with ψ
=
α
φ
2
′ on D
3. One obtains α
φ
1
-
φ
2
′
=
0 on D
2 where, by applying the inductive hypothesis, we infer that there exists a form χ
∈
Λ
i
-
2
σ
,
U
D
2
,
X, such that φ
1
-
φ
2
′
=
α
ψ. We will retain from Taylor’s decomposition of χ on D
2 a sufficient number of terms, such that χ
′ (the retained part) verifies α
χ
-
α
χ
′
<
1
/
2 on D
-
1. Thinking analogously, we can define a sequence of forms φ
ν, φ
ν
∈
Λ
i
-
1
σ
,
U
D
ν
+
1
,
X, enjoying the properties
(18)
ψ
=
α
φ
ν
W
on
⁡
D
ν
+
1
,
φ
ν
+
1
-
φ
ν
<
1
2
ν
+
1
W
on
⁡
D
.
The sequenceφ
ν obviously converges to a form having the analytic coefficients on U and satisfying ψ
=
α
φ on U.The uniqueness of the spectralS-capacities for S-decomposable operators systems can be proved. We will now prove this on other ways, emphasizing the connection between the spectral S-capacity related to an operator and certain linear subspaces, described using the local spectrum, which is most useful.Leta be a commuting system of operators on the space X, a
⊂
B
X, with the residual spectrum S
a. If H is an arbitrary set from C
n such that H
⊃
S
a, we will put X
a
H
=
x
,
x
∈
X
,
sp
a
,
x
⊂
H and X
a
H
=
x
,
x
∈
X
,
σ
a
,
x
⊂
H; X
a
H and X
a
H are linear subspaces of X and X
a
H
⊂
X
a
H.Theorem 9.
Ifa
=
a
1
,
a
2
,
…
,
a
n
⊂
B
X is an S-decomposable system and E is its S-spectral capacity, then
(19)
E
F
=
X
a
F
,
for any closed set F
⊂
C
n, F
⊃
S.Proof.
According to Theorem8, S
⊃
S
a; hence F
⊃
S
a and X
a
F makes sense. The inclusion E
F
⊂
X
a
F follows by the fact that sp
a
,
x
⊂
σ
a
,
E
F (Proposition 2 (4°)). In the same manner as for [8, Theorem 2.2.1], one proves the inverse inclusion, with the observation that F is not an arbitrary subset of C
n, but F
⊃
S.Corollary 10.
Let a be anS-decomposable system. Then for any closed F
⊃
S, the subspace X
a
F is spectral maximal space of a; more precisely, for any subspace Z invariant to a such that σ
a
,
Z
⊂
F, one has Z
⊂
X
a
(
F
); moreover σ
a
,
X
a
F
⊂
F.Proof.
The inclusionσ
a
,
X
a
F
⊂
F follows by the preceding theorem, since σ
a
,
E
F
⊂
F.
IfZ is invariant to a with σ
a
,
Z
⊂
F then any z
∈
Z
sp
a
,
z
⊂
σ
a
,
Z
⊂
F; hence z
∈
X
a
F, meaning Z
⊂
X
a
(
F
).Proposition 11.
If a isS-decomposable then, for any x
∈
X, one has s
p
a
,
x
=
σ
a
,
x.Proof.
Let us prove first thatsp
a
,
x
⊂
σ
a
,
x or its equivalent σ
a
,
x
⊂
r
a
,
x. Let z
∈
δ
a
,
x and according to [8, Theorem  1.1.3.] let us consider an open neighbourhood V of z and n analytic functions defined on V taking values in X, f
1
,
f
2
,
…
,
f
n that verify the equality
(20)
x
=
ζ
1
-
a
1
f
1
ζ
+
⋯
+
ζ
n
-
a
n
f
n
ζ
,
ζ
∈
V. We consider the n
-
1 degree form defined on V,
(21)
ψ
ζ
=
∑
i
=
1
n
-
1
i
-
1
f
i
ζ
s
1
∧
s
2
∧
⋯
∧
s
i
∧
⋯
∧
s
n
.
This form can be considered as an element ofΛ
n
-
1
σ
∪
d
z
-
,
C
∞
V
,
X and it easily verifies the equality s
x
=
α
⊕
∂
-
ψ on V taking into account the analyticity of the functions f
i
(
δ
f
i
=
0
); hence it results that V
⊂
d
a
,
x; that is, δ
a
,
x
⊂
d
a
,
x or g
a
,
x
⊂
γ
a
,
x hence g
a
,
x
∪
S
a
=
sp
a
,
x
⊂
γ
a
,
x
∪
S
a
=
σ
a
,
x. For the inverse inclusion σ
a
,
x
⊂
sp
a
,
x, let z
∈
r
a
,
x and let D be an open polydisk with its centre in z such that D
⊂
r
a
,
x.
Sincex
∈
X
a
(
sp
a
,
x
=
E
(
sp
a
,
x
)
), hence by [8, Theorem 1.1.3.] there exist the analytic functions f
1
,
f
2
,
…
,
f
n defined on D and taking values in X, such that
(22)
x
=
∑
i
=
1
n
ζ
i
-
a
i
f
i
ζ
,
ζ
∈
D
.
That means thatz
∈
ρ
a
,
x; hence r
a
,
x
⊂
ρ
a
,
x hence σ
a
,
x
⊂
sp
a
,
x.Corollary 12.
If a is anS-decomposable system, then for any H
⊂
C
n with H
⊃
S one has X
a
H
=
X
a
H.Proof.
It easily follows by the preceding proposition.Proposition 13.
If a is an arbitrary system of operators, then(23)
σ
a
,
X
=
⋃
x
∈
X
s
p
a
,
x
.Proof.
The inclusion⋃
x
∈
X
sp
a
,
x
⊂
σ
a
,
X results from Proposition 2 (4°),  sp
a
,
x
⊂
σ
a
,
X.
Conversely, ifz
∈
⋂
x
∈
X
r
a
,
x, then H
n
U
z
,
X
,
α
=
0; since z
∈
S
a, there exists an open polydisk D, with D
∩
S
a
=
Ø, for which, according to Theorem 8, we have H
i
U
D
,
X
,
α
=
0
(
i
=
0,1
,
…
,
n
-
1
).
Then by [8, Corollary 1.4.3.] it follows that z
∈
r
a
,
x; hence ⋃
x
∈
X
sp
a
,
x
⊃
σ
a
,
X.Definition 14.
The support of an S-spectral capacity E is the set
(24)
supp
E
=
∩
F
;
F
closed
,
E
F
=
X
.Proposition 15.
Ifa
=
a
1
,
a
2
,
…
,
a
n
⊂
B
X is an S-decomposable system and E is its S-spectral capacity, then supp
⁡
E
=
σ
a
,
X.Proof.
The inclusionσ
a
,
X
⊂
supp
E results from the fact that, for any closed F such that E
F
=
X, we have σ
a
,
X
=
σ
a
,
E
F
⊂
F, where σ
a
,
X
⊂
∩
F
,
F
closed
,
E
F
=
X
=
supp
E. For the inverse inclusion, let z
0
∈
r
a
,
X and let us prove that z
0
∉
supp
E. Let V be an open neighborhood of z
0 such that V
-
⊂
r
a
,
X and let F be a closed set under the conditions z
0
∉
F, F
⊃
S and X
=
E
F
+
E
V
-; these conditions are possible because z
0
∈
S
(
S
⊂
σ
a
,
X
).
Letx
∈
E
V
-; since V
⊂
r
a
,
X it results that s
x
=
α
⊕
∂
-
ψ in a neighborhood C
n
∖
V of the spectrum sp
a
,
X; hence by applying [8, formula 1.2.4], we find x
=
0. Therefore E
V
-
=
0; that is, E
F
=
X; hence from z
0
∉
F it follows that z
0
∉
supp
E.Corollary 16.
Ifa
=
a
1
,
a
2
,
…
,
a
n
⊂
B
X is an S-decomposable system, then one has σ
a
,
E
F
⊂
σ
a
,
X, for any closed set F
⊂
C
n.Proof.
We have(25)
E
F
∩
σ
a
,
X
=
E
F
∩
E
σ
a
,
X
=
E
F
∩
X
=
E
F
,
hence
(26)
σ
a
,
E
F
=
σ
a
,
E
F
∩
σ
a
,
X
⊂
F
∩
σ
a
,
X
⊂
σ
a
,
X
.


## 4. Conclusions

The originality consists in the adaptation of the results obtained for the spectral capacities by Frunză to the decomposable systems of operators to theS-spectral capacities which are appropriate to the S-decomposable systems of operators.These theorems remain valid also for a special class ofS-decomposable systems, namely, S-spectral systems. An important result was obtained in the particular case when the topological dimension of S is zero and the S-decomposable (S-spectral) systems become decomposable (spectral systems).We will underline the relevance, importance, and necessity of studying theS-decomposable operators showing the consistence of this class, in the sense of how many and how substantial its subfamilies are. Relations with this same subject can also be found in the “regularities” of Müller [11]. The S-decomposable operators appear firstly as restrictions and quotients of decomposable operators, particularly restrictions and quotients of spectral operators, and therefore A
S-scalar or S-spectral operators (generalized S-spectral, generalized S-scalar, etc.).



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