Title

Anti-G-hermiticity preserving linear maps that preserve strongly the invertibility of Hilbert space operators

Date of Publication

2017

Document Type

Dissertation

Degree Name

Doctor of Philosophy in Mathematics

Subject Categories

Mathematics

College

College of Science

Department/Unit

Mathematics and Statistics Department

Thesis Adviser

Jose Tristan F. Reyes

Defense Panel Chair

Edmundo R. Perez Jr.

Defense Panel Member

Isagani B. Jos
Julius Cesar C. Agapito
Job A. Nable

Abstract/Summary

A linear map : A ! B of algebras A and B preserves strong invertibility if (x{u100000}1) = (x){u100000}1 for all x 2 A{u100000}1, where A{u100000}1 denotes the set of invertible elements of A. Let B(H) be the Banach algebra of all bounded linear operators on a separable complex Hilbert space H with dimH = 1. An operator U 2 B(H) is said to be anti-G-Hermitian if U] = {u100000}U, where U] denotes the G-adjoint of U. A linear map : B(H) ! B(H) preserves anti-G-Hermiticity if (U)] = {u100000} (U) for every anti-G- Hermitian operator U on H. In this paper, we characterize a continuous unital linear map : B(H) ! B(H) that preserves anti-G-Hermiticity and preserves strongly the invertibility of Hilbert space operators. The discussion is in the context of G- operators, that is, linear operators on H with respect to a xed but arbitrary positive de nite Hermitian operator G 2 B(H){u100000}1. From the Hilbert space H with an inner product h i, we consider a new inner product [ ] in H such that [x y] = hGx yi for all x y 2 H. We present a discussion of operators on (H [ ]) analogous to the discussion of operators on (H h i).

The discussion of operators on (H [ ]) will be extended to the quotient algebra of B(H) by the ideal K(H) of compact operators on H, which is known as the Calkin algebra C(H) of operators on H. We present the anti-G-Hermiticity and invertibility preserving properties of the canonical map : B(H) ! C(H). We then introduce vii the continuous unital linear map : C(H) ! C(H) induced by the linear map : B(H) ! B(H) which preserves essentially anti-G-Hermiticity and preserves strongly the invertibility of operators on H. We also take a look at the preserving properties and the characterization of the induced map.

Abstract Format

html

Language

English

Format

Electronic

Accession Number

CDTG007170

Shelf Location

Archives, The Learning Commons, 12F Henry Sy Sr. Hall

Physical Description

1 computer disc; 4 3/4 in.

Keywords

Linear operators

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