by Hai-Wei Sun. / Thesis (Ph.D.)--Chinese University of Hong Kong, 1996. / Includes bibliographical references. / Abstracts --- p.1 / Summary --- p.3 / Introduction --- p.3 / Summary of the papers A-C --- p.5 / Paper A --- p.19 / Paper B --- p.34 / Paper C --- p.60

Typescript (photocopy) / 2 v. : ill. ; 30 cm. / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / Thesis (Ph.D.)--University of Adelaide, Dept. of Computer Science, University of Adelaide, 1982

Let $vp$ be a $L^infty$ function on the unit circle $Bbb T$ and $ au$ be an elliptic automorphism on the unit disc $Bbb D$. In this paper, we will show that $T_vp C_ au$, i.e., the product of the Toeplitz operator $T_vp$ and the composition operator $C_ au$ on $H^2$, is similar to a block Toeplitz matrix if $ au$ has finite order.

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.

published_or_final_version / Mathematics / Master / Master of Philosophy

published_or_final_version / Mathematics / Master / Master of Philosophy

Thesis (Ph.D.) -- University of Adelaide, Dept. of Computer Science, University of Adelaide, 1982. / Typescript (photocopy).

Thesis (M. Phil.)--University of Hong Kong, 2000. / Includes bibliographical references (leaves 51-53).

Thesis (M. Phil.)--University of Hong Kong, 1995. / Includes bibliographical references (leaf 53-54).

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.