## Electronic Theses and Dissertations

#### Date

2021

#### Date of Award

2021

#### Document Type

Dissertation

#### Degree Name

Doctor of Philosophy

#### Committee Chair

Fernanda Botelho

#### Committee Member

John Haddock

#### Committee Member

Anna Kaminska

#### Committee Member

Irena Lasiecka

#### Abstract

The focus of this dissertation is two-folded. First, it deals with a thorough study of hermitian operators and hermitian projections on spaces of bounded operators and Banach spaces at large. Second, it concerns extensions and reinterpretations of results dealing with the characterization of geometric aspects of Banach spaces. This includesthe characterization of extreme points of the unit ball of tensor product spaces and their impact on the form of the surjective isometries supported by those spaces.The dissertation is divided into 4 chapters. The first chapter is the introduction, the second chapter describes the hermitian operators on several settings of spaces of continuous and integrable functions and on spaces of bounded operators. We also consider operators which are both Hermitian as well as projections. Characterizations of such operators is done for spaces of vector valued continuous functions and for spaces of bounded operators. In particular, for two Banach spaces X and Y under certain conditions, the characterizations of Hermitian projections is as follows. Let (X,Y) be an ideal pair of Banach spaces. Let P : B(X, Y) to B(X, Y). Then P is a Hermitian projection if and only if either there exists a Hermitian projection Q in B(X) such that P(A)=AQ for every A in B(X, Y), or there exists an Hermitian projection R in B(Y) such that P(A)=RA for every A in B(X, Y). Chapter 3 deals with the study of the extreme points on spaces of tensor products. We follow the approached given in the paper by Stegall and Ruess and extend their result for the tensor product of complex Banach spaces. The main point is the characterization of the extreme points of the injective tensor product X tensor Y but a description for the projective tensors are also derived. The knowledge of the extreme points is very important in the derivation of the form for the surjective isometries. In chapter 4 we investigate the theorem due to Jarosz for the characterization of isometries on the injective tensor product space X tensor Y under some assumptions on the component spaces X and Y. The last point of this work deals with the characterization of generalized bicircular projection on the injective tensor product X and K with the assumption that K is strictly convex.

#### Library Comment

Dissertation or thesis originally submitted to ProQuest

#### Recommended Citation

Dey, Priyadarshi, "Hermitian projections and geometric properties of Banach spaces" (2021). *Electronic Theses and Dissertations*. 2514.

https://digitalcommons.memphis.edu/etd/2514

## Comments

Data is provided by the student.