# Subnormal operators with compact selfcommutator

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## 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. 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.

## Details

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### Subject classification (UKÄ) – MANDATORY

• Mathematics
Original language English 353-367 Manuscripta Mathematica 91 1 Published - 1996 Research Yes Yes