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Toeplitz operators with piecewise continuous symbols on the Hardy space $H^1$ (1808.01788v2)
Published 6 Aug 2018 in math.FA, math.CV, and math.SP
Abstract: The geometric descriptions of the (essential) spectra of Toeplitz operators with piecewise continuous symbols are among the most beautiful results about Toeplitz operators on Hardy spaces $Hp$ with $1<p<\infty$. In the Hardy space $H1$, the essential spectra of Toeplitz operators are known for continuous symbols and symbols in the Douglas algebra $C+H\infty$. It is natural to ask whether the theory for piecewise continuous symbols can also be extended to $H1$. We answer this question in negative and show in particular that the Toeplitz operator is never bounded on $H1$ if its symbol has a jump discontinuity.