Berezin Transforms And Toeplitz Operators

Toeplitz operators are of importance in connection with a variety of problems infunction theory, differential equation, Von Neumann algebra, noncommutativegeometry, random matrix, information and control theory, quantum mechanics, andseveral other fields. Our study on the Toeplitz operator and Toeplitz algebra plays animportant role in the development of mathematics, Physics, Engineering andTechnology. In this thesis, we study the positivity and invertibility of Toeplitzoperators on the Bergman space via Berezin transforms. We divide the dissertation intosix chapters:In Chaper1, we mainly introduce the background and development of theproblems in this thesis, and then state our main results.In the second chapter, we introduce the main tools uesd to study our problems:Berezin transform, n-th Berezin transform and reproducing kernels. We introduce thedefinition and the properties of the Berezin transform, we also state the relationshipsbetween the compactness and boundedness of the Toeplitz operators on Bergman space(or Hardy space) and the Berezin transforms. In this chaper we will see that manyproblems on Toeplitz operators may become easy when we handle them by Berezinthransform and n-th Berezin transform.In the third chapter, we devote to study the positivity of Toeplitz operators onBergman space by using Berezin transform. We want to study the relationships betweenthe positivity of Bergman Toeplitz operators and their Berezin transforms. Indeed, onone hand, we use Nazarov’s results to show that the positivity of Toeplitz operators onthe Bergman space is not completely determined by the positivity of the Berezintransform of their symbols; on the other hand, we consider the radial function (z) a z2b z c a, b,c on the unit disk and provide a sufficient andnecessary condition on the positivity of T by its matrix. Using the Berezin transformof the function, we obtain the necessary condition for to be nonnegative on theunit disk, and we prove that the necessary condition on the positivity of BergmanToeplitzs is not0. Applying the Sturm theorem, we show that there exists a familyof functions such that their Berezin transforms are positive and bounded below on theunit disk but the Toeplitz operators with these symbols are not positive.In Chapter4, our main consideration is the relationships between the invertibility of Bergman Toeplitz operators with harmonic symbols and their Berezin transforms.Our research is motivated by the following Douglas’s question:If is a bounded function on and the harmonic extension of is boundedbelow on the diak, then is the Toeplitz operator T invertible on the Hardy space?We expect to establish the sufficient or necessary condition on the invertibility ofBergman Toeplitz operators by the Berezin transform and n-th Berezin transform.Using Luecking’s characterization on the invertibility of Toeplitz operators with positivesymbols, we are able to give a equivalent condition on the invertibility of this type ofoperators by the Berezin transform. Furthermore, for the Toeplitz operators withsymbolsh (z) az bz c a, b,c,Taking lot mileage, we are able to prove thatTh is invertible if and only if h isbounded below on the disk, more importantly, we learn about the spectrum of this typeof operators. For a bounded harmonic function on, we obtain a sufficient conditionfor the Toeplitz operators with this symbol to be invertible by means of the eatimationof the norm of Hankel operators. This condition is analogous to the Chang-Tolokonnikov result for Hardy Toeplitz operators. For the general case, we also give asufficient condition on the invertibility of Toeplitz operators with bounded symobols viaBerezin transform and n-th Berezin transform.In Chapter5, using Berezin transform, we study the positivity of Toeplitz operatorson the harmonic Bergman spacesL2h ()andL2h (). We mainly consider the Toeplitzoperators with boubded harmonic symbols in these spaces, and we construct a class ofToeplitz operators such that they are not positive but the Berezin transforms of theirsymbols are positive and bounded below on the unit disk.In the final chapter, we summarize the main results of the thesis, pose the difficultswhich we have not overcome and introduce some new problems which we will study inthe future.