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On the Berezin-Toeplitz calculus

Author: L. A. Coburn
Journal: Proc. Amer. Math. Soc. 129 (2001), 3331-3338
MSC (2000): Primary 47B35; Secondary 47B32
Published electronically: March 29, 2001
MathSciNet review: 1845010
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Abstract: We consider the problem of composing Berezin-Toeplitz operators on the Hilbert space of Gaussian square-integrable entire functions on complex $n$-space, $\mathbf{C}^n$. For several interesting algebras of functions on $\mathbf{C}^n$, we have $T_\varphi T_\psi =T_{\varphi \diamond \psi }$for all $\varphi ,\psi $ in the algebra, where $T_\varphi $ is the Berezin-Toeplitz operator associated with $\varphi $ and $\varphi \diamond \psi $ is a ``twisted'' associative product on the algebra of functions. On the other hand, there is a $C^\infty $ function $\varphi $ for which $T_\varphi $ is bounded but $T_\varphi T_\varphi \neq T_\psi $ for any $\psi$.

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Additional Information

L. A. Coburn
Affiliation: Department of Mathematics, State University of New York at Buffalo, Buffalo, New York 14260

Received by editor(s): December 21, 1999
Received by editor(s) in revised form: March 21, 2000
Published electronically: March 29, 2001
Additional Notes: The author’s research was supported by a grant of the NSF and a visiting membership in the Erwin Schrödinger Institute.
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2001 American Mathematical Society

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