Some Properties Of Block Toeplitz Operators In Vector Valued Function Spaces

Block Toeplitz operators arise naturally in several fields of mathematics, in a variety of problems in physics and in quantum mechanics. The hyponormality of Toeplitz operators is closely related to Halmos’s Problem5in his lecture "Ten problems in Hilbert spaces ":Is every subnormal Toeplitz operator either normal or analytic? The definition of subnormality implies hyponormality, there is "Bram-Halmos " criterion in the converse implication.In Chapter1, we briefly introduce the background and basic information of this thesis, current status of related research at home and aborad.In Chapter2, we reduce the problem of two block Toeplitz operator product to the problem of finite sum of two Toeplitz operator product on the scalar Bergman space and use the matrix characteristics, we get some explicit results about the finite rank semicommutators, commutators and zero product of two block Toeplitz operators with harmonic symbols on the vector valued Bergman space. Especially, we get the results analogous to the scalar case.In Chapter3, we focus on the hyponormality of block Toeplitz operators with matrix valued harmonic symbols on the vector valued Bergman space. By the matrix structure and using function theory, we get some necessary conditions and sufficient conditions for the hyponormal block Toeplitz operators.In Chapter4, we study the hyponormality of Toeplitz operators on the harmonic Dirichlet space. Based on the decomposition of the Sobolev space, using the matrix representation of both the Toeplitz operators and the adjoint of the Toeplitz operators acting on {Zk}k≥0, we completely characterize the hyponormality of bounded Toeplitz operators with Sobolev symbols W1,∞(D) on the Dirichlet space. Similar methods applied in the harmonic Dirichlet space, we get the similar results. Especially, the bounded Toeplitz operators with harmonic symbols on the Dirichlet space or the harmonic Dirichlet space are hyponormal if and only if they are constants.