Hyponormality and spectra of Toeplitz operators

Farenick, Douglas; Lee, Woo

doi:10.1090/S0002-9947-96-01683-2

Hyponormality and spectra of Toeplitz operators
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by Douglas R. Farenick and Woo Young Lee PDF

Trans. Amer. Math. Soc. 348 (1996), 4153-4174 Request permission

Abstract:

This paper concerns algebraic and spectral properties of Toeplitz operators $T_{\varphi }$, on the Hardy space $H^{2}({\mathbb {T}})$, under certain assumptions concerning the symbols $\varphi \in L^{\infty }({\mathbb {T}})$. Among our algebraic results is a characterisation of normal Toeplitz opertors with polynomial symbols, and a characterisation of hyponormal Toeplitz operators with polynomial symbols of a prescribed form. The results on the spectrum are as follows. It is shown that by restricting the spectrum, a set-valued function, to the set of all Toeplitz operators, the spectrum is continuous at $T_{\varphi }$, for each quasicontinuous $\varphi$. Secondly, we examine under what conditions a classic theorem of H. Weyl, which has extensions to hyponormal and Toeplitz operators, holds for all analytic functions of a single Toeplitz operator with continuous symbol.

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