Toeplitz operators with piecewise continuous symbols – a neverending story?

*(English)*Zbl 0849.47013This article is a slightly modified and English version of a plenary talk given at the DMV-Jahrestagung in Duisburg, 1994.

The purpose of the article is to present a few pieces of the fascinating development in the theory of Toeplitz operators with piecewise continuous symbols. We start with Toeplitz’ 1911 paper, review the classical \(L^2\) results obtained until the middle of the sixties, embark on the \(L^p\) theory worked out by Gohberg and Krupnik in the seventies, cite the 1990 result by Spitkovsky on operators in spaces with Muckenhoupt weights, and end up with phenomena for operators on Carleson curves which were discovered by Karlovich and the author only in 1994. Although with the consideration of Muckenhoupt weights and Carleson curves the theory has now reached a certain maximal level of generality, an end of the metamorphosis of the spectra of Toeplitz operators with piecewise continuous symbols doesn’t seem to be in sight.

The purpose of the article is to present a few pieces of the fascinating development in the theory of Toeplitz operators with piecewise continuous symbols. We start with Toeplitz’ 1911 paper, review the classical \(L^2\) results obtained until the middle of the sixties, embark on the \(L^p\) theory worked out by Gohberg and Krupnik in the seventies, cite the 1990 result by Spitkovsky on operators in spaces with Muckenhoupt weights, and end up with phenomena for operators on Carleson curves which were discovered by Karlovich and the author only in 1994. Although with the consideration of Muckenhoupt weights and Carleson curves the theory has now reached a certain maximal level of generality, an end of the metamorphosis of the spectra of Toeplitz operators with piecewise continuous symbols doesn’t seem to be in sight.