The diagonal entries in the formula “quasitriangular - compact = triangular” and restrictions of quasitriangularity
Authors: Herrero Domingo A. 
Published: July 2010

A (bounded linear) Hilbert space operator

T
T

is called quasitriangular if there exists an increasing sequence

{

P
n

}

n
=
0

∞

\{ {P_n}\} _{n = 0}^\infty

of finite-rank orthogonal projections, converging strongly to 1, such that

‖

(
1
−

P
n

)
T

P
n

‖

→
0

(
n
→
∞
)

\left \| {(1 - {P_n})T{P_n}} \right \| \to 0\,(n \to \infty )

. This definition, due to P. R. Halmos, plays a very important role in operator theory. The core of this article is a concrete answer to the following problem: Suppose

T
T

is a quasitriangular operator and

Γ
=
{

λ
j

}

j
=
1

∞

\Gamma = \{ {\lambda _j}\} _{j = 1}^\infty

is a sequence of complex numbers. Find necessary and sufficient conditions for the existence of a compact operator

K
K

(of arbitrarily small norm) so that

T
−
K

T - K

is triangular with respect to some orthonormal basis, and the sequence of diagonal entries of

T
−
K

T - K

coincides with

Γ
\Gamma

. For instance, if no restrictions are put on the norm of

K
K

, then

T
T

and

Γ
\Gamma

must be related as follows: (a) if

λ
0

{\lambda _0}

is a limit point of

Γ
\Gamma

and

λ
0

−
T

{\lambda _0} - T

is semi-Fredholm, then

ind

(

λ
0

−
T
)
>
0

{\operatorname {ind}}({\lambda _0} - T) > 0

; and (b) if

Ω
\Omega

is an open set intersecting the Weyl spectrum of

T
T

, whose boundary does not intersect this set, then

{
j
:

λ
j

∈
Ω
}

\{ j:{\lambda _j} \in \Omega \}

is a denumerable set of indices. Particularly important is the case when

Γ
=
{
0
,
0
,
0
,
…
}

\Gamma = \{ 0,0,0, \ldots \}

. The following are equivalent for an operator

T
T

: (1) there is an integral sequence

{

P
n

}

n
=
0

∞

\{ {P_n}\} _{n = 0}^\infty

of orthogonal projections, with rank

P
n

=
n

{P_n} = n

for all

n
n

, converging strongly to 1, such that

‖

(
1
−

P
n

)
T

P

n
+
1

‖

→
0

(
n
→
∞
)

\left \| {(1 - {P_n})T{P_{n + 1}}} \right \| \to 0\,(n \to \infty )

; (2) from some compact

K
,

T
−
K

K,\,T - K

is triangular, with diagonal entries equal to 0; (3)

T
T

is quasitriangular, and the Weyl spectrum of

T
T

is connected and contains the origin. The family

(

StrQT

)

−
1

{({\text {StrQT}})_{ - 1}}

of all operators satisfying (1) (and hence (2) and (3)) is a (norm) closed subset of the algebra of all operators; moreover,

(

StrQT

)

−
1

{({\text {StrQT}})_{ - 1}}

is invariant under similarity and compact perturbations and behaves in many senses as an analog of Halmos’s class of quasitriangular operators, or an analog of the class of extended quasitriangular operators

(

StrQT

)

−
1

{({\text {StrQT}})_{ - 1}}

, introduced by the author in a previous article. If

{

P
n

}

n
=
0

∞

\{ {P_n}\} _{n = 0}^\infty

is as in (1), but condition

‖

(
1
−

P
n

)
T

P

n
+
1

‖

→
0

(
n
→
∞
)

\left \| {(1 - {P_n})T{P_{n + 1}}} \right \| \to 0\,(n \to \infty )

is replaced by (1’)
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar left-parenthesis 1 minus upper P Subscript n Sub Subscript k Subscript Baseline right-parenthesis upper T upper P Subscript n Sub Subscript k Subscript plus 1 Baseline double-vertical-bar right-arrow 0 left-parenthesis k right-arrow normal infinity right-parenthesis">

‖

(
1
−

P

n
k

)
T

P

n
k

+
1

‖

→
0

(
k
→
∞
)

\left \| {(1 - {P_{{n_k}}})T{P_{{n_k} + 1}}} \right \| \to 0\,(k \to \infty )

for some subsequence

{

n
k

}

k
=
1

∞

\{ {n_k}\} _{k = 1}^\infty

, then (1’) is equivalent to (3’),

T
T

is quasitriangular, and its Weyl spectrum contains the origin. The family

(

QT

)

−
1

{({\text {QT}})_{ - 1}}

of all operators satisfying (1’) (and hence (3’)) is also a closed subset, invariant under similarity and compact perturbations, and provides a different analog to Halmos’s class of quasitriangular operators. Both classes have “

m
m

-versions” (

(

StrQT

)

−
m

{({\text {StrQT}})_{ - m}}

and, respectively,

(

QT

)

−
m

{({\text {QT}})_{ - m}}

,

m
=
1
,
2
,
3
,
…

m = 1,2,3, \ldots

) with similar properties. (

(

StrQT

)

−
m

{({\text {StrQT}})_{ - m}}

is the class naturally associated with triangular operators

A
A

such that the main diagonal and the first

(
m
−
1
)

(m - 1)

superdiagonals are identically zero, etc.) The article also includes some applications of the main result to certain nest algebras “generated by orthonormal bases.”