Compactness And Schatten Class Of Toeplitz-type Operators

Analytic function space theory is closely related to other branches of mathematics,such as harmonic analysis,quantum theory,partial differential equations and so on.Operators theory on analytic function space provides methods for these fields.Analytic function space is a blend of complex function theorem with harmonic analysis and operator theory.Therefore,the study of function space theory is indispensable for the development of contemporary mathematics.The operator theory on analytic function space mainly studies the properties of operators by using the properties of symbols.This paper mainly studies some properties of Toeplitz-type operators on Bergman space and Fock space,For example,boundedness,compactness and Schatten p-class.The paper is organized as follows:In the first chapter,we introduce background of the operator theory on analytic function spaces.We give some properties of Toeplitz operators.Meanwhile,we give the definition of Toeplitz-type operators on Bergman space and Fock space.In Chapter 2,on Bergman space,we give a necessary and sufficient condition for Tμ(j)to be in the Schatten p-class for 1 ≤p<∞ on the Bergman sapce A2 by using average function and Berezin transform of symbol,and then we give a sufficient condition for Tμ(j)to be in the Schatten p-class(0<p<1)on A2 by using average function of lattice.We give the equivalent condition that the finite sum of the finite product of Toeplitz-type operators induced by bounded symbols on Bergman space Ap(1<p<∞)is a compact operator.In Chapter 3,On Fock space,we characterize that the boundedness and compactness of Toeplitz-type operators are characterized by using the properties of Berezin transformation of symbol.And for 1≤P<∞,we give the necessary and sufficient conditions for Toeplitz-type operators to belong to Schatten p-class.In Chapter 4,We give some necessary and some sufficient conditions for invertibility of general Toeplitz operators on the Fock space.In particular,we study the Fredholm properties of Toeplitz operators with BMO1 symbols,which their Berezin transform are bounded functions of vanishing oscillation.We prove that the Toeplitz operator with such a symbol is Fredholm on Fock space if and only if the Berezin transform of symbol is bounded away from zero outside a sufficiently large disk in the complex plane.We also show that the Fredholm index of the Toeplitz operator can be computed via the winding of it’s Brerezin transform along a sufficiently large circle and give a characterization of invertible Toeplitz operators with nonnegative symbols,possibly unbounded,and such that the Berezin transform of symbols are bounded and of vanishing oscillation.