Journal of Operator Theory

Volume 53, Issue 2, Spring 2005 pp. 431-440.

Compact Toeplitz operators with unbounded symbols

Authors: Joseph A. Cima (1) and \v{Z}eljko \v{C}u\v{c}kovi\'{c} (2)
Author institution: (1) Department of Mathematics, University of North Carolina, Chapel Hill, NC 27599-3250, USA
(2) Department of Mathematics, The University of Toledo, Toledo, OH 43606-3390, USA

Summary: We construct bounded Toeplitz operators on the Bergman space

$L^2_a$ on the unit disk, whose symbols are unbounded functions. These operators can be compact and in some cases Hilbert-Schmidt. In fact we show that for any (essentially unbounded) function

$H \in L^2$ there is a set

$\Gamma$ in the unit disk such that the (essentially unbounded) function given by

$h=\chi_{\Gamma}H$ is the symbol for a compact Toeplitz operator on

$L^2_a$ .

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