Spectral properties of slant Toeplitz operators
Abstract
Let $\varphi(\theta) \sim \sum\limits\sbsp{-\infty}{\infty}\ c\sb{n}e\sp{in\theta}$ be an $L\sp\infty$ function on the unit circle T. An operator $A\sb{\varphi}$ on $L\sp2$(T) is called a slant Toeplitz operator with symbol $\varphi$ if it satisfies $\langle A\sb{\varphi}e\sp{im\theta}, e\sp{in\theta}\rangle$ = $c\sb{2n-m}$ for all n, m in $\doubz$. In the beginning of this thesis, we will show that all slant Toeplitz operators can be written in the form $A\sb1M\sb{\varphi}$, where $M\sb{\varphi}$ is the multiplication operator on $L\sp2$(T), and from that, we will show that the adjoint of a slant Toeplitz operator is a weighted composition operator. Then, using these facts, we will study basic properties of the slant Toeplitz operators such as norms, eigenspaces, spectra and essential spectra as well as some structural properties of the $C\sp*$ algebra generated by these operators. We will also demonstrate how to use the Perron-Frobenius type theory for the "positive" operators on ordered Banach spaces and techniques from the theory of composition operators on functional Banach spaces to derive some of the major results in this thesis about the spectra of slant Toeplitz operators with continuous symbols. Furthermore, we will present a numerical technique to compute the spectral radii of those slant Toeplitz operators whose symbols are trigonometric polynomials.
Degree
Ph.D.
Advisors
Cowen, Purdue University.
Subject Area
Mathematics
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