Given a function (more generally, a measure) on a locally compact Abelian group, one can define the Toeplitz operators as certain integral transforms of functions on the dual group, where the kernel is the Fourier transform of the original function or measure. In the case of the unit circle, this corresponds to forming a matrix out of the Fourier coefficients in a particular way. We will study the asymptotic eigenvalue distributions of these Toeplitz operators.

Let £f be a complex number in the closed unit disk D , And H be a separable Hilbert space with the orthonormal basis , say ,£`= {e_n:n=0,1,2,¡K}. A bounded operator T on H is called a £f- Toeplitz operator if <Te_(n+1) ,e_(m+1) >=£f<Te_n ,e_m > (where < , > is inner product on H) The L^2 function £p~ £Ua_n e^in£c with a_n=<Te_0 ,e_n> for n>=0 , and a_n=<Te_n ,e_0 > for n<0 is , on the other hand , called the symbol of T The subject arises naturally from a special case of the operator equation S^* AS=£fA+B where S is a shift on H , which plays an essential role in finding bounded matrix (a_ij ) on L^2 (Z) that solves the system of equations {((a_(2i,2j) =p_ij+aa_ij@a_(2i,2j-1) =q_ij+ba_ij )@a_(2i-1,2j) =£h_ij+ca_ij@a_(2i-1,2j-1) =£s_ij+da_ij ) ¢t, for all i ,j belong Z , where (p_ij ) ,(q_ij ) ,(£h_ij ) ,(£s_ij ) are bounded matrices on l^2 (Z) and a ,b ,c ,d belong C . It is also clear that the well-known Toeplitz operators are precisely the solutions of S^* AS=A , when S is the unilateral shift . In this paper , we will determine the spectra of £f- Toeplitz operators with |£f|=1 of finite order, and when the symbols are analytic with C^1 boundary values.

Thesis (M.A.)--Wake Forest University. Dept. of Mathematics, 2008. / Vita. Includes bibliographical references (leaves 48-49)

Dissertation (Ph.D.)--University of Toledo, 2009. / Typescript. "Submitted as partial fulfillment of the requirements for the Doctor of Philosophy Degree in Mathematics." Bibliography: leaves 57-59.

In this work we examine C*-algebras of Toeplitz operators over the unit ball in ℂⁿ and the unit polydisc in ℂ². Toeplitz operators are interesting examples of non-normal operators that generate non-commutative C*-algebras. Moreover, in the nice cases (depending on the geometry of the domain) of algebras of Toeplitz operators we can recover some analogues of the spectral theorem up to compact operators. In this setting, we can capture the index of a Fredholm operator which is a fundamental numerical invariant in Operator Theory. / Master of Science

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We prove a Fedosov index formula for the index of Toeplitz operators connected with the Hardy space of solutions to an elliptic system of first order partial differential equations in a bounded domain of Euclidean space with infinitely differentiable boundary.

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In this work we investigate multi-variable Toeplitz operators and their relationship with KK-theory in order to apply this relationship to define and analyze embedded Toeplitz problems. In particular, we study the embedded Toeplitz problem of the unit disk into the unit ball in C^2. The embedding of Toeplitz problems suggests a way to define Toeplitz operators over singular spaces. / Ph. D.

This work is an introduction to anisotropic spaces, which have an ω-weight of analytic functions and are generalizations of Lipshitz classes in the polydisc. We prove that these classes form an algebra and are invariant with respect to monomial multiplication. These operators are bounded in these (Lipshitz and Djrbashian) spaces. As an application, we show a theorem about the division by good-inner functions in the mentioned classes is proved.