A dense set of operators with tiny commutants
Authors: Herrero Domingo A. 
Published: June 2012

For a (bounded linear) operator

T
T

on a complex, separable, infinite-dimensional Hilbert space

H

\mathcal {H}

, let

A

(
T
)

\mathcal {A}\,(T)

and

A

a

(
T
)

{\mathcal {A}^a}(T)

denote the weak closure of the polynomials in

T
T

and, respectively, the weak closure of the rational functions with poles outside the spectrum of

T
T

. Let

A

′

(
T
)

\mathcal {A}’(T)

and

A

(
T
)

\mathcal {A}(T)

denote the commutant and, respectively, the double commutant of

T
T

. We say that

T
T

has a tiny commutant if

A

′

(
T
)
=

A

a

(
T
)

\mathcal {A}’(T)= {\mathcal {A}^a}(T)

. By constructing a large family of "models" and by using standard techniques of approximation, it is shown that

T
∈

L

(

H

)
:
T

T \in \mathcal {L}\,(\mathcal {H}):T

has a tiny commutant is norm-dense in the algebra

L

(

H

)

\mathcal {L}\,(\mathcal {H})

of all operators acting on

H

\mathcal {H}

. Other related results: Let

Lat

B

\operatorname {Lat}\;\mathcal {B}

denote the invariant subspace lattice of a subalgebra

B

\mathcal {B}

of

L

(

H

)

\mathcal {L}(\mathcal {H})

. For a Jordan curve

γ
⊂

C

\gamma \subset {\mathbf {C}}

, let

γ
^

\hat \gamma

denote the union of

γ
\gamma

and its interior; for

T
∈

L

(

H

)

T \in \mathcal {L}\;(\mathcal {H})

, let

ρ

s
−
F

(
T
)
=
{
λ
∈

C

:
λ
−
T

{\rho _{s - F}}\,(T)= \{ \lambda \in {\mathbf {C}}:\lambda - T

is a semi-Fredholm operator, and let
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="rho Subscript s minus upper F Superscript plus Baseline left-parenthesis upper T right-parenthesis left-parenthesis rho Subscript s minus upper F Superscript minus Baseline left-parenthesis upper T right-parenthesis right-parenthesis equals left-brace lamda element-of rho Subscript s minus upper F Baseline left-parenthesis upper T right-parenthesis colon ind left-parenthesis lamda minus upper T right-parenthesis greater-than 0 left-parenthesis greater-than 0 comma resp period right-parenthesis right-brace">

ρ

s
−
F

+

(
T
)
(

ρ

s
−
F

−

(
T
)
)
=
{
λ
∈

ρ

s
−
F

(
T
)
:

ind

(
λ
−
T
)
>
0

(
>
0
,

resp.)} 

\rho _{s - F}^ + (T)(\rho _{s - F}^ - (T))= \{ \lambda \in {\rho _{s - F}}(T):{\text {ind}}(\lambda - T) > 0\;(> 0,{\text {resp.)\} }}

. With this notation in mind, it is shown that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-brace upper T element-of script upper L left-parenthesis script upper H right-parenthesis colon script upper A left-parenthesis upper T right-parenthesis equals script upper A Superscript a Baseline left-parenthesis upper T right-parenthesis right-brace Superscript minus Baseline equals left-brace upper T element-of script upper L left-parenthesis script upper H right-parenthesis colon upper L a t script upper A left-parenthesis upper T right-parenthesis equals upper L a t script upper A Superscript a Baseline left-parenthesis upper T right-parenthesis right-brace Superscript minus Baseline equals left-brace upper A element-of script upper L left-parenthesis script upper H right-parenthesis">

{
T
∈

L

(

H

)
:

A

(
T
)
=

A

a

(
T
)

}
−

=

{
T
∈

L

(

H

)
:
Lat

A

(
T
)
=
Lat

A

a

(
T
)

}
−

=
{
A
∈

L

(

H

)

{\{ T \in \mathcal {L}(\mathcal {H}):\mathcal {A}(T)= {\mathcal {A}^a}(T)\} ^ - } = {\{ T \in \mathcal {L}(\mathcal {H}):\operatorname {Lat}\;\mathcal {A}(T)= \operatorname {Lat}\;{\mathcal {A}^a}(T)\} ^ - }= \{ A \in \mathcal {L}(\mathcal {H})

if

γ
\gamma

(Jordan curve)

⊂

ρ

s
−
F

±

(
A
)

\subset \rho _{s - F}^ \pm (A)

, then

γ
^

⊂
σ
(
A
)
}

\hat \gamma \subset \sigma (A)\}

; moreover,

{
A
∈

L

(

H

)

\{ A \in \mathcal {L}(\mathcal {H})

: if

γ
\gamma

(Jordan curve)

⊂

ρ

s
−
F

±

(
A
)

\subset \rho _{s - F}^ \pm (A)

, then

ind

(
λ
−
A
)

{\text {ind}}(\lambda - A)

is constant on
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove gamma With caret intersection rho Subscript s minus upper F Baseline left-parenthesis upper A right-parenthesis right-brace subset-of left-brace upper T element-of script upper L left-parenthesis script upper H right-parenthesis colon script upper A left-parenthesis upper T right-parenthesis equals script upper A prime left-parenthesis upper T right-parenthesis right-brace Superscript minus Baseline subset-of StartSet upper T element-of script upper L left-parenthesis script upper H right-parenthesis colon upper L a t script upper A left-parenthesis upper T right-parenthesis equals upper L a t script upper A prime left-parenthesis upper T right-parenthesis EndSet subset-of left-brace upper A element-of script upper L left-parenthesis script upper H right-parenthesis">

γ
^

∩

ρ

s
−
F

(
A
)
}
⊂

{
T
∈

L

(

H

)
:

A

(
T
)
=

A

′

(
T
)

}
−

⊂
{
T
∈

L

(

H

)
:
Lat

A

(
T
)
=
Lat

A

′

(
T
)
}
⊂
{
A
∈

L

(

H

)

\hat \gamma \cap {\rho _{s - F}}(A)\} \subset {\{ T \in \mathcal {L}(\mathcal {H}):\mathcal {A}(T)= \mathcal {A}’(T)\} ^ - } \subset \{ T \in \mathcal {L}(\mathcal {H}):\operatorname {Lat}\;\mathcal {A}(T)= \operatorname {Lat}\;\mathcal {A}’(T)\} \subset \{ A \in \mathcal {L}(\mathcal {H})

: if

γ
\gamma

(Jordan curve)

⊂

ρ

s
−
F

±

(
A
)

\subset \rho _{s - F}^ \pm (A)

, then
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove gamma With caret intersection rho Subscript s minus upper F Baseline left-parenthesis upper A right-parenthesis subset-of rho Subscript s minus upper F Superscript plus-or-minus Baseline left-parenthesis upper A right-parenthesis right-brace subset-of StartSet upper T element-of script upper L left-parenthesis script upper H right-parenthesis colon script upper A left-parenthesis upper T right-parenthesis equals script upper A Superscript a Baseline left-parenthesis upper T right-parenthesis EndSet">

γ
^

∩

ρ

s
−
F

(
A
)
⊂

ρ

s
−
F

±

(
A
)
}
⊂
{
T
∈

L

(

H

)
:

A

(
T
)
=

A

a

(
T
)
}

\hat \gamma \cap {\rho _{s - F}}(A) \subset \rho _{s - F}^ \pm (A)\} \subset \{ T \in \mathcal {L}(\mathcal {H}):\mathcal {A}(T)= {\mathcal {A}^a}(T)\}

. (The first and the last inclusions are proper.) The results also include a partial analysis of

Lat

A

(
T
)

\operatorname {Lat}\;\mathcal {A}(T)

.