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### Subnormal operators with compact selfcommutator

(1996) In Manuscripta Mathematica 91(1). p.353-367
Abstract
If \$S\$ is a hyponormal operator, then Putnam's inequality gives an estimate on the norm of the self-commutator \$[S^*,S]\$, while the Berger-Shaw theorem gives (under appropriate cyclicity hypotheses) a corresponding estimate on the trace of \$[S^*,S]\$. Of course these results hold when \$S\$ is subnormal.

In the subnormal setting, the author obtains useful estimates on the norm and essential norm of commutators of the form\break \$[T_u,S]\$, where \$T_u\$ is a Toeplitz operator with continuous symbol \$u\$. A consequence is the following compactness condition. If the essential spectrum of \$S\$ is the boundary of an open set, then \$[S^*,S]\$ is compact.

The author also proves some trace estimates for commutators.... (More)
If \$S\$ is a hyponormal operator, then Putnam's inequality gives an estimate on the norm of the self-commutator \$[S^*,S]\$, while the Berger-Shaw theorem gives (under appropriate cyclicity hypotheses) a corresponding estimate on the trace of \$[S^*,S]\$. Of course these results hold when \$S\$ is subnormal.

In the subnormal setting, the author obtains useful estimates on the norm and essential norm of commutators of the form\break \$[T_u,S]\$, where \$T_u\$ is a Toeplitz operator with continuous symbol \$u\$. A consequence is the following compactness condition. If the essential spectrum of \$S\$ is the boundary of an open set, then \$[S^*,S]\$ is compact.

The author also proves some trace estimates for commutators. His basic method is a careful analysis of positive operator-valued measures.

Abstract For an arbitrary subnormal operator we estimate the essential norm and trace of commutators of the form [T u, S], whereT u is a Toeplitz operator with continuous symbol. In particular, we obtain criteria for the compactness of [S *,S]. The trace estimates apply to multiplication operators on Hardy spaces over general domains. (Less)
author
publishing date
type
Contribution to journal
publication status
published
subject
in
Manuscripta Mathematica
volume
91
issue
1
pages
353 - 367
publisher
Springer
ISSN
1432-1785
DOI
10.1007/BF02567960
language
English
LU publication?
no
id
05ff9680-6f1b-4205-a714-feb3a7f8d6a9 (old id 1467218)
2016-04-01 11:50:06
date last changed
2020-05-04 04:29:25
```@article{05ff9680-6f1b-4205-a714-feb3a7f8d6a9,
abstract     = {If \$S\$ is a hyponormal operator, then Putnam's inequality gives an estimate on the norm of the self-commutator \$[S^*,S]\$, while the Berger-Shaw theorem gives (under appropriate cyclicity hypotheses) a corresponding estimate on the trace of \$[S^*,S]\$. Of course these results hold when \$S\$ is subnormal. <br/><br>
<br/><br>
In the subnormal setting, the author obtains useful estimates on the norm and essential norm of commutators of the form\break \$[T_u,S]\$, where \$T_u\$ is a Toeplitz operator with continuous symbol \$u\$. A consequence is the following compactness condition. If the essential spectrum of \$S\$ is the boundary of an open set, then \$[S^*,S]\$ is compact. <br/><br>
<br/><br>
The author also proves some trace estimates for commutators. His basic method is a careful analysis of positive operator-valued measures.<br/><br>
<br/><br>
Abstract For an arbitrary subnormal operator we estimate the essential norm and trace of commutators of the form [T u, S], whereT u is a Toeplitz operator with continuous symbol. In particular, we obtain criteria for the compactness of [S *,S]. The trace estimates apply to multiplication operators on Hardy spaces over general domains.},
author       = {Aleman, Alexandru},
issn         = {1432-1785},
language     = {eng},
number       = {1},
pages        = {353--367},
publisher    = {Springer},
series       = {Manuscripta Mathematica},
title        = {Subnormal operators with compact selfcommutator},
url          = {http://dx.doi.org/10.1007/BF02567960},
doi          = {10.1007/BF02567960},
volume       = {91},
year         = {1996},
}

```