# On spectral analysis of self-adjoint Toeplitz operators

## TL;DR

This paper extends the spectral analysis of self-adjoint Toeplitz operators, improving existing methods, exploring finite spectral multiplicity, and developing detailed analysis for piecewise continuous symbols to aid scattering theory.

## Contribution

It provides an improved proof of absolute continuity, introduces generalized eigenfunctions, and advances spectral analysis for piecewise continuous symbols.

## Key findings

- Enhanced proof of absolute continuity using a weak limiting absorption principle
- Introduction of generalized eigenfunctions for finite spectral multiplicity
- Detailed spectral analysis for piecewise continuous symbols

## Abstract

The paper pursues three objectives. Firstly, we provide an expanded version of 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.