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.

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Research areas and keywords

Subject classification (UKÄ) – MANDATORY

  • Mathematics
Original languageEnglish
Pages (from-to)353-367
JournalManuscripta Mathematica
Volume91
Issue number1
Publication statusPublished - 1996
Publication categoryResearch
Peer-reviewedYes
Externally publishedYes