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Proceedings of the American Mathematical Society

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Toeplitz operators on Cartan domains essentially commute with a bilateral shift

Author: Miroslav Engliš
Journal: Proc. Amer. Math. Soc. 117 (1993), 365-368
MSC: Primary 47B35; Secondary 32M15
MathSciNet review: 1123651
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Abstract: For bounded symmetric domains $ \Omega \subset {{\mathbf{C}}^N}$, a bilaterial shift operator $ U$ is shown to exist on the Bergman space $ {A^2}(\Omega )$ such that $ U{T_f} - {T_f}U$ is a compact operator for all Toeplitz operators $ {T_f}$. This may be viewed as an extension of the well-known fact that $ {S^{\ast}}TS - T = 0$ whenever $ T$ is a Toeplitz operator on $ {H^2},\;S$ being the unilateral shift. It also follows that the $ {C^{\ast}}$-algebra generated by Toeplitz operators on $ {A^2}(\Omega )$ does not contain all bounded operators.

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Keywords: Toeplitz operators, bounded symmetric domains
Article copyright: © Copyright 1993 American Mathematical Society

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