## Arts & Sciences Electronic Theses and Dissertations

#### Title

Some Problems in Reproducing Kernel Spaces

#### ORCID

http://orcid.org/0000-0002-0736-0987

Spring 5-15-2022

#### Author's School

Graduate School of Arts and Sciences

Mathematics

#### Degree Name

Doctor of Philosophy (PhD)

Dissertation

#### Abstract

The two chapters of this thesis are comprised of work in the setting of reproducing kernel (Hilbert) spaces. These are Banach (or Hilbert) spaces of functions defined on some set, with the special property that point evaluation, on the underlying set, is bounded.

The first chapter deals with the study of inner functions. These functions have a rich history in function and operator theory in the Hardy spaces of the unit disk. The first section of this chapter studies the relationship between generalized inner functions and optimal polynomial approximants. The second section, which is joint work with Trieu Le, deals with a generalization of a classical type of inner function (finite Blaschke product). The last section, which is joint work with Raymond Cheng, considers the (Banach) space $\ell^p_A$-- the space of analytic functions on the disk with $p$-summable Maclaurin coefficients. We consider the geometry of the multiplier algebra of this space and characterize extremal multipliers.

The second chapter considers the geometry of two planar sets associated to linear operators acting on reproducing kernel Hilbert spaces. The first section of this chapter, which is joint work with Carl Cowen, considers the convexity of the Berezin range of an operator on a reproducing kernel Hilbert space. We focus primarily on a class of composition operators acting on the Hardy space of the unit disk. The final section of the chapter, and the thesis, joint work with Benjamin Russo and Douglas Pfeffer, deals with the connectedness of various spectra of certain Toeplitz acting on a family of sub-Hardy Hilbert spaces.

English (en)

John E. McCarthy

Carl C. Cowen

COinS

#### DOI

https://doi.org/10.7936/7dx2-px16