Putnam's Inequality and Analytic Content in the Bergman Space

• Main Author: Fleeman, Matthew
• Format: Others
• Published: Scholar Commons 2016
• Subjects: Bergman spaces; Operator theory; Approximation theory; Mathematics
• Online Access: http://scholarcommons.usf.edu/etd/6238; http://scholarcommons.usf.edu/cgi/viewcontent.cgi?article=7434&context=etd

In this dissertation we are interested in studying two extremal problems in the Bergman space. The topics are divided into three chapters. In Chapter 2, we study Putnam’s inequality in the Bergman space setting. In [32], the authors showed that Putnam’s inequality for the norm of self-commutators can be improved by a factor of 1 for Toeplitz operators with analytic symbol φ acting on the Bergman space A2(Ω). This improved upper bound is sharp when φ(Ω) is a disk. We show that disks are the only domains for which the upper bound is attained. In Chapter 3, we consider the problem of finding the best approximation to z ̄ in the Bergman space A2(Ω). We show that this best approximation is the derivative of the solution to the Dirichlet problem on ∂Ω with data |z|2 and give examples of domains where the best approximation is a polynomial, or a rational function. Finally, in Chapter 4 we study Bergman analytic content, which measures the L2(Ω)-distance between z ̄ and the Bergman space A2(Ω). We compute the Bergman analytic content of simply connected quadrature domains with quadrature formula supported at one point, and we also determine the function f ∈ A2(Ω) that best approximates z ̄. We show that, for simply connected domains, the square of Bergman analytic content is equal to torsional rigidity from classical elasticity theory, while for multiply connected domains these two domain constants are not equivalent in general.