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Journal of Operator Theory

Volume 84, Issue 2, Fall 2020  pp. 453-485.

On spectral analysis of self-adjoint Toeplitz operators

Authors:  Alexander V. Sobolev (1), Dmitri Yafaev (2)
Author institution: (1) UCL Department of Mathematics, Gower Str., London WC1E 6BT, U.K.
(2) University Rennes, CNRS, IRMAR-UMR 6625, F-35000 Rennes, France and SPGU, Univ. Nab. 7/9, Saint Petersburg, 199034 Russia

Summary: The paper pursues three objectives. First, we provide an expanded version of the spectral analysis of self-adjoint Toeplitz operators, initially built by M. Rosenblum in the 1960's. We offer some improvements to Rosenblum's approach: for instance, our proof of the absolute continuity, relying on a weak version of the limiting absorption principle, is more direct. Secondly, we study in detail Toeplitz operators with finite spectral multiplicity. In particular, we introduce generalized eigenfunctions and investigate their properties. Thirdly, we develop a more detailed spectral analysis for piecewise continuous symbols. This is necessary for construction of scattering theory for Toeplitz operators with such symbols.

Keywords: Toeplitz operators, spectral decomposition, discontinuous symbols

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