Journal of Operator Theory

Volume 49, Issue 2, Spring 2003 pp. 325-346.

Toeplitz operators on the unit ball in $\scriptstyle \C^n$ with radial symbols

Authors: S. Grudsky, (1) A. Karapetyants, (2) and N. Vasilevski (3)
Author institution: (1) Departamento de Matematicas, CINVESTAV del I.P.N., Apartado Postal 14--740, 07000 Mexico, D.F. Mexico
(2) Department of Mathematics, 344711 Rostov-on-Don, Russia
(3) Departamento de Matematicas, CINVESTAV del I.P.N., Apartado Postal 14--740, 07000 Mexico, D.F. Mexico

Summary: The paper is devoted to the study of Toeplitz operators with radial symbols on the weighted Bergman spaces on the unit ball in $\C^n$. Admitting ``badly" behaved unbounded symbols we get new qualitative features. In particular, contrary to known results, a Toeplitz operator with the same (unbounded) symbol now can be bounded in one weighted Bergman space and unbounded in another, compact in one weighted Bergman space and bounded but not compact in another, compact in one weighted Bergman space and unbounded in another. In our case of radial symbols, the Wick (or covariant) symbol of a Toeplitz operator gives complete information about the operator, providing its spectral decomposition.

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