The Hyponormal And M-hyponormal Toeplitz Operators On The Bergman Space And The Fock Space

Operator theory in function spaces has always been one of the important branches of functional analysis.It mainly studies the properties of operators in various function spaces.Bergman space and Fock space are very important function spaces which have received wide attention.Toeplitz operator is the most well-known and widely studied operator in function spaces.And the hyponormality and M-hyponormality of Toeplitz operators have always been hot topics for scholars to study.Through extensive literature,there are relatively few studies on hyponormal Toeplitz operators on the Fock space and most of them focus on harmonic polynomial symbolic functions.The hyponormality of Toeplitz operators with non-harmonic functions is less characterized.So far,most of the research on M-hyponormal operators is to study its properties on Hilbert space.However,there seems to be little research on the symbolic characteristics of M-hyponormal Toeplitz operator.On the basis of previous studies,we characterize the hyponormal and M-hyponormal Toeplitz operators on the Bergman space and the Fock space.This paper is organized as follows.In Chapter 1,the research development history of Toeplitz operators is briefly described.Then the research progress and some achievements of hyponormal Toeplitz operators on different function spaces are introduced.Finally,the research status of M-hyponormal operators on Hilbert spaces are stated.In Chapter 2,the basic knowledge related to this thesis is introduced.In Chapter 3,we study the hyponormal Toeplitz operators on the Fock space with some non-harmonic symbols.Consider?(z)=zⁿ+?|z|^s,where??C,n?N,s?(0,+?).We show that the necessary and sufficient condition for the hyponormality of T_?is?=0,when1?n?2[s/2].And a further example is given to illustrate that the conclusion is not valid if the assumption1?n?2[s/2] is removed.Next,we find the necessary conditions and sufficient conditions for the hyponormality of T_?and T_?in terms of the zeros of symbolic functions,where?(z)=(?)z(z+?)and?(z)=(?)(z-?₁)(z+?₁),respectively.In Chapter 4,we characterize the M-hyponormality of some Toeplitz operators on the Bergman space,the weighted Bergman space and the Fock space,respectively.In Chapter 5,the conclusion of this thesis is summarized and prospected.