Classes of unbounded Toeplitz operators

Abstract

The theory of bounded Toeplitz operators with symbols in L∞(T) have been studied extensively in the last century with numerous sources describing their Fredholm and spectral theory, invertibility, their kernels and their adjoints. Once we consider symbols beyond L∞(T), the respective Toeplitz operators become unbounded and their theory becomes complicated. In recent literature, various definitions have surfaced, which yield unbounded Toeplitz operators with more general symbols, which appear to be dichotomous and display faint correlation. In this dissertation, we study some of the recent approaches to unbounded Toeplitz operators and provide connections between them. Moreover, in some cases, we can describe their Fredholm and spectral properties and their adjoints, so as to give an extension of the results from the well-known bounded case.