On the essential norms of Toeplitz operators with continuous symbols

DOI10.1016/J.JFA.2020.108835arXiv2007.13178MaRDI QIDQ6345868

Eugene Shargorodsky

Publication date: 26 July 2020

Abstract: It is well known that the essential norm of a Toeplitz operator on the Hardy space $H^p(mathbbT)$, $1<p<infty$ is greater than or equal to the $L^{i}nfty(mathbbT)$ norm of its symbol. In 1988, A. B"ottcher, N. Krupnik, and B. Silbermann posed a question on whether or not the equality holds in the case of continuous symbols. We answer this question in the negative. On the other hand, we show that the essential norm of a Toeplitz operator with a continuous symbol is less than or equal to twice the $L^{i}nfty(mathbbT)$ norm of the symbol and prove more precise $p$-dependent estimates.

Mathematics Subject Classification ID

Norms (inequalities, more than one norm, etc.) of linear operators (47A30) Toeplitz operators, Hankel operators, Wiener-Hopf operators (47B35) Spaces of operators; tensor products; approximation properties (46B28) Measures of noncompactness and condensing mappings, (K)-set contractions, etc. (47H08)

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