Subnormal block Toeplitz operators

Abstract

Let H be a complex Hilbert space and B(H) denotes the algebra of all bounded linear operators on H. An operator T ∈ B(H) is said to be normal if T ∗ T = T T ∗ , hyponormal if T ∗ T ≥ T T ∗ and subnormal if it has a normal extension, where T ∗ denotes the adjoint of T. P. R. Halmos posed a problem, known as Halmos’ Problem 5, concerning the characterization of subnormal Toeplitz operators on the Hardy space. The problem asks: “Is every subnormal Toeplitz operator either normal or analytic?” C. Cowen answered this question negatively, leading to the refined problem: “Which subnormal Toeplitz operators are either normal or analytic?” In this thesis, we address the refined version of Halmos’ Problem 5, particularly in the context of Toeplitz and block Toeplitz operators with finite rank self-commutator and answer a problem recently posed by R. E. Curto, I. S. Hwang and W. Y. Lee [22, Problem 6.2]. We establish conditions for identifying subnormal block Toeplitz operators whose self-commutators are of finite rank. Furthermore, we investigate the subnormality and hyponormality of Toeplitz operators with operator-valued symbols and present sufficient conditions for hyponormality in this setting. In addition, we provide characterizations of hyponormality and subnormality for analytic Toeplitz operators on the Hardy space of Hilbert space-valued functions.