Toeplitz Operators On Spaces Of Functions

Toeplitz operator theory on function spaces is an important topic in functional anal-ysis. Toeplitz operators have much close relation with many branches of mathematicsand physics, such as, function theory, operator theory, cybernetics, quantum mechanics,probability, partial diferential equation, harmonic analysis, computational mathematics,etc. Research on Toeplitz operator properties has become a primary area of modern math-ematics and has obtained abundant results. It has formed a series of theoretical systemafter a long time of study.People mainly focus on the studies of boundedness, compactness, Fredholm proper-ties, algebra properties, spectra properties, essential norm, invariant subspaces of Toeplitzoperators on function spaces. In1911, Toeplitz firstly started his research on Toeplitzmatrices. People gradually expand it to some kinds of operator theory of function spaces,such as, Toeplitz operator theory on Hardy space, Bergman space, Dirichlet space. Alot of important relative results were obtained. So many mathematicians contribute toToeplitz operator theory, such as, Sz¨ego, Hartman, Wintner, R.G.Douglas, A.B¨ottcher,H. Upmeier, K.H.Zhu, S. Axler, D.C.Zheng, G.F.Cao.In this thesis, we discuss the Schatten-p class properties of Toeplitz operators onDirichlet space and pluri-harmonic Bergman space of unit ball inCn, characterize theboundedness, compactness, Fredholm properties of Toeplitz operators on Dirichlet space.By Berezin type tranform, the compactness and problem of invariant subspaces, invert-ibility of Toeplitz operators on Dirichlet space are studied.In chapter one, we introduce some information about research on Toeplitz operator,the main results and the content of this thesis. In chapter two, we construct Schatten p-class (0<p <∞) Toeplitz operators on Dirichlet space with symbol functions which areunbounded on any neighborhood of each boundary point of unit ballBninCn. In chapterthree, we construct Schatten p-class (0<p <∞) Toeplitz operators on pluri-harmonicspace with symbol functions which are unbounded on any neighborhood of each boundarypoint of unit ballBninCn. In chapter four, we research the boundedness, compactness,Fredholm properties of Toeplitz operators on Dirichlet space Dp, compute Fredholm index.In chapter five, we research the boundedness, compactness, Fredholm properties of Toeplitzoperators on Dirichlet space D1, compute Fredholm index. In chapter six, we discuss therelationship between the compactness of Toeplitz operators on Dirichlet space D2and its Berezin transform. In chapter seven, by Berezin type tranform, we discuss the compactnessand problem of invariant subspaces, invertibility of Toeplitz operators on Dirichlet space.