Abstract Let £f be a complex number in the closed unit disc , and H be a separable Hilbert space with the orthonormal basis, say,£`= {en : n =0 , 1 , 2¡K}. A bounded operator T on H is called a £f-Toeplitz operator if <Tem+1 , en+1> =£f <Tem , en> (where <¡E,¡E> is the inner product on H).If the function £p can be represented as a linear combination of the above orthonormal basis with the coefficients an=<Te0 ,en >, n≥ 0,and an=<Telnl ,e0 >, n<0, then we call this 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 , and in this operator equation the matrix A can solve a special set of simultaneous equations. 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 review the similarities and differences between £f-Toeplitz operators and Toeplitz operators. The main purpose is to generalize the well-known Coburn's characterization for the essential spectrum(or,the same in this case,spectrum)for Toeplitz operators to £f-Toeplitz operators.

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Let £f be a complex number in the closed unit disc, 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_{m+1},Te_{n+1} >=£f< Te_m,Te_n > (where <¡E,¡E> is the inner product on H). The subject arises just recently 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} = v_{ij} + ca_{ij} a_{2i−1,2j−1} = w_{ij} + da_{ij} for all i, j ∈ Z, where (p_{ij}), (q_{ij}), (v_{ij}), (w_{ij}) are bounded matrices on l^2(Z) and a, b, c, d ∈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. The purpose of this paper is to discuss some basic topics, such as boundedness and compactness, of the £f-Toeplitz operators, and study the similarities and the differences with the corresponding results for the Toeplitz operators.

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Thesis (Ph.D.)--George Mason University, 2009. / Vita: p. 158. Thesis director: Flavia Colonna. Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics. Title from PDF t.p. (viewed Oct. 11, 2009). Includes bibliographical references (p. 150-157). Also issued in print.

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Bochum, Univ., Diss., 2003. / Computerdatei im Fernzugriff.