Dual Toeplitz Operators And Products Of Hankel Operators In The Several Complex Variables

Operator theory in function spaces has being the hot problem for discussion in the field of operator theory and analysis, and it is closely related with operator theory, operator algebra, function theory, differential equation, complex analysis, differential topology and so on. On the other hand, it has many important applications in control, quantum mechanics, probability and statistics, etc. Toeplitz operators and Hankel operators, the most important operators on the function spaces, which have the extremely profound affect to the operator theory, the operator algebra and the complex analysis, have attracted the attentions of many scholars.Since 1950s, the enthusiasms of the mathematicians for the research of Toeplitz operators and Hankel operators have been escalated continuously, and many important achievements have been obtained [1-7]. Since most of the studies are based on the Hardy space and the Bergman space of the unit disk, there is still a lack of their properties on the higher dimensional region for the reason that the topological boundary of the higher dimensional region is very complicated and the multi-dimensional complex analysis theory is very whimsical, which make it difficult to be extended. This thesis is mainly about to discuss the commutativity and the algebra properties of dual Toeplitz operators, and to prove the boundedness and compactness of products of Hankel operators in the several complex variables. This thesis is organized as follows:Some background information about Toeplitz operators, dual Toplitz operators and Hankel operators is reviewed and several operators products are introduced in Chapter 1, including Toeplitz products, Hankel products, Haplitz products and so on.The second chapter mainly deals with commutativity of dual Toeplitz operators of the polydisk, such as the characterizations of commuting dual Toeplitz operators, essentially commuting dual Toeplitz operators and essentially semi-commuting dual Toeplitz operators. The following results are presented: By the property of mean value of the holomorphic functions and the property of the reproducing kernel k_ω, we get the rank one operator k_ω(?) k_ωby a finite sum of finite products of Toeplitz operators on L_α²(Dⁿ), and define the operator L_ω. Under this condition, we describe when two dual Toeplitz operators with bounded measurable functions symbol commute. As a consequence, we make a conclusion that the dual Toeplitz operator is normal if and only if the range of the symbol lies on a line. Moreover, we discuss the characterizations of essentially commuting dual Toeplitz operators and essentially semi-commuting dual Toeplitz operators.Chapter 3 discusses the algebraic and spectral properties of dual Toeplitz operators on the orthogonal complement of the Bergman space, such as boundedness, compactness, spectral properties and so on. Brown and Halmos [3] showed that the only compact Toeplitz operator on the Hardy space is the zero operator and a Toeplitz operator is bounded on the Hardy space if and only if its symbol is bounded. This is easily seen to be false for the Toeplitz operator on the Bergman space. As a matter of fact, there are so many unbounded symbols that induce bounded Toeplitz operators. We will prove that the only compact dual Toeplitz operator is the zero operator, and that a densely defined dual Toeplitz operator with square integrable symbol is bounded if and only if its symbol is essentially bounded. We construct a symbol map on the dual Toeplitz algebra generated by all bounded dual Toeplitz operators. As an application of our symbol map we obtain a necessary condition on symbols of a finite number of dual Toeplitz operators whose product is the zero operator. Finally, we discuss spectral properties of dual Toeplitz operators. We prove a spectral inclusion and give examples to show that in general the spectrum and essential spectrum of a dual Toeplitz operator can be disconnected.Products of Toeplitz operators and Hankel operators on the Bergman space of the polydisk are investigated in Chapter 4. Utilizing the expression of k_ω(?) k_ω, we consider the question for which square integrable function f and g on the polydisk the densely defined products H_fH_g^* are bounded on the Bergman space, and a necessary condition and a sufficient condition are obtained. We prove that a condition slightly stronger than the necessary condition we get is sufficient for the boundedness of Hanekl operators. Furthermore, we obtain similar results for mixed Haplitz products T_fH_g^* and H_gT_f-. Finally, the necessary and sufficient conditions for compactness of Hankel products are discussed.Products of Toeplitz operators and Hankel operators on the weighted Bergman space of the unit disk form the topic of Chapter 5. The weighted Bergman space A_α² is the space of analytic functions on D which are square integrable with respect to the measure dA_α(z) = (α+ 1)(1-|z|²)^αdA(z). The functions k_ω^(α) are the normalized reproducing kernels for A_α². We also found a expression of rank one operator k_ω^(α)(?)k_ω^(α) on the weighted Bergman space. Especially, whenαis a nonnegative integer, it is just a finite sum of finite products of Toeplitz operators. Using this expression, we get the characterizations of the boundedness of operators products, similar to the case of the unweighted. Finally, we discuss the compactness of operators products when a is a nonnegative integer, which is also similar to the case of the unweighted.