Some contributions to Toeplitz operators on the Bergman spaces

Abstract

In this thesis, we are concerned with Toeplitz operators acting on the Bergman spaces of the unit ball in the n-dimensional complex space $C^n$. The main purpose of the thesis is on the size estimate of the Toeplitz operators. More precisely, we mainly study the relationship between operator theoretic properties of Toeplitz operators and function theoretic properties of their symbols. In Chapter 1, we consider the commuting problem for Toeplitz operators acting on the $L^2$-Bergman space. In the one dimensional case, Axler and Cuckovic has recently showed that two Toeplitz operators with bounded harmonic symbols commute only in the obvious case. In this chapter, we consider the same problem with bounded pluriharmonic symbols on the ball and partially extend the Axler and Cuckovic result to the ball. In Chapter 2, we consider the characterization problem of compact Toeplitz operators on the $L^2$-Bergman space of a product of balls rather than the ball. In the ball or the polydisk setting, Zheng has recently characterized bounded symbols of compact Toeplitz operators in terms of certain boundary vanishing properties. In this chapter, we use a new argument to extend Zheng``s result to general product of balls, and obtain a new characterization and show that a certain restriction in Zheng``s characterization is inessential. In Chapter 3, we introduce a Banach space of holomorphic functions on the ball motivated by the atomic decompositions in the sense of Luecking, and establish its dual and predual spaces. At the same time, we describe these spaces in terms of boundedness, compactness and membership in the Schatten p-classes of certain Toeplitz operators acting on the $L^2$-Bergman space of the ball. In Chapter 4, we consider a spectral property of Toeplitz operators acting on $L^p$-Bergman space of the ball. In one dimensional setting, Zeng has considered a symbol continuous up to boundary and computed the essential spectrum of the corresponding Toeplitz ope...