# 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

## 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

## Recommended Citation

Buscano, J. D. (2017). Anti-G-hermiticity preserving linear maps that preserve strongly the invertibility of Hilbert space operators. Retrieved from https://animorepository.dlsu.edu.ph/etd_doctoral/509