published_or_final_version / Mathematics / Master / Master of Philosophy

Thesis (M. Phil.)--University of Hong Kong, 1994. / Includes bibliographical references (leaves 47-49).

published_or_final_version / Mathematics / Master / Master of Philosophy

published_or_final_version / Mathematics / Master / Master of Philosophy

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. Phil.)--University of Hong Kong, 2002. / Includes bibliographical references (leaves 51-52).

published_or_final_version / abstract / toc / Mathematics / Master / Master of Philosophy

Thesis (M. Phil.)--University of Hong Kong, 2008. / Includes bibliographical references (leaf 45-46) Also available in print.

Thesis (M. Phil.)--University of Hong Kong, 2009. / Includes bibliographical references (leaf 49) Also available in print.

Let X be a compact Hausdorff topological space. The Banach space C(X) consists of all continuous complex value functions with the supnorm. An operator P on C(X) is called a generalized bicircular projection if P + £f(I − P) is an isometry for all |£f| = 1, £f in C and P2 = P. In this thesis, we study some projections which are the averages of two composition operators or two weighted composition operators on C(X). If a projection is the average of the identity and a composition operator, it is a generalized bicircular projection. And give an example of a projection which is the average of the identity and a weighted composition operator, but not a generalized bicircular projection. We also discuss some projections which are the average of two bounded linear operators on a Banach space. And the main result is that, let T1 and T2 are two bounded linear operators on a Banach space, and Q = T1+T2 2 . If T1 ¡CT2 = T2 ¡CT1 and T2 1 = T2 2 = Id then Q is a tripotent, i.e. Q3 = Q.