Toeplitz Operators On Reproducing Kernel Hilbert Spaces

In this thesis,we mainly discuss Toeplitz operators with essentially bounded function symbols on Bergman type spaces and Toeplitz operators with finite posi-tive Borel measure symbols on weighted Bergman spaces.Using Berezin transform-s,we obtain some necessary conditions of the solvability of the Riccati equations and the existence of invariant subspaces for Toeplitz operators with essentially bounded function symbols on these Bergman type spaces.Using Berezin trans-forms and Carleson measures,we characterize the boundedness and compactness of Toeplitz operators with finite positive Borel measure symbols on these weighted Bergman spaces whose weights satisfying₂conditions.In Chapter 1,we introduce the backgrounds about research on Toeplitz op-erators as well as the main results and the main content of this thesis.In Chapter 2,we introduce some basic tools which will be used in our research,the basic definitions and properties of Bergman type spaces and weighted Bergman spaces.In Chapter 3,we study Toeplitz operators with essentially bounded function symbols on Bergman type spaces.Using Berezin transforms,we obtain some necessary conditions of the solvability of the Riccati equations and the existence of invariant subspaces for these Toeplitz operators.In addition,a class of special operator equations is considered.In Chapter 4,we study Toeplitz operators with finite positive Borel measure symbols on weighted Bergman spaces whose weights satisfying₂condition.Using the reproducing kernel functions,we obtain the atomic decomposition for these weighted Bergman spaces.Using Berezin transforms and Carleson measures,we characterize under some conditions the boundedness and compactness of these Toeplitz operators.