Operators on Reproducing Kernel Hilbert Spaces

Abstract

The thesis contains outcomes of research on operators on reproducing kernel Hilbert spaces. The most well-known classes of reproducing kernel Hilbert spaces are Bergman space, Hardy space, Dirichlet space, and Fock space, etc. In general, the theories of these reproducing kernel Hilbert spaces are used in prediction theory, quantum mechanics, functional analysis, complex analysis, and operator theory, etc., but the focus of the present study is on the Bergman space and its operators. Primarily, the study focuses on bounded linear operators like Toeplitz and little Hankel operators on the Bergman space. Despite several studies conducted on Toeplitz and little Hankel operators on Bergman space, few questions remain unaddressed. This thesis addresses the algebraic and operator-theoretic properties of these operators on the Bergman space. In particular, normality, positivity, isometry, boundedness, unitariness, range inclusion relationship, operator inequalities and equalities, and finite rank of Toeplitz and little Hankel operators on the Bergman space are studied. newline