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

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...

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Main Author: Buscano, Jay D.
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Language:English
Published: Animo Repository 2017
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Online Access:https://animorepository.dlsu.edu.ph/etd_doctoral/509
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spelling oai:animorepository.dlsu.edu.ph:etd_doctoral-15082024-08-06T02:11:51Z Anti-G-hermiticity preserving linear maps that preserve strongly the invertibility of Hilbert space operators Buscano, Jay D. 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. 2017-01-01T08:00:00Z text https://animorepository.dlsu.edu.ph/etd_doctoral/509 Dissertations English Animo Repository Linear operators Mathematics
institution De La Salle University
building De La Salle University Library
continent Asia
country Philippines
Philippines
content_provider De La Salle University Library
collection DLSU Institutional Repository
language English
topic Linear operators
Mathematics
spellingShingle Linear operators
Mathematics
Buscano, Jay D.
Anti-G-hermiticity preserving linear maps that preserve strongly the invertibility of Hilbert space operators
description 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.
format text
author Buscano, Jay D.
author_facet Buscano, Jay D.
author_sort Buscano, Jay D.
title Anti-G-hermiticity preserving linear maps that preserve strongly the invertibility of Hilbert space operators
title_short Anti-G-hermiticity preserving linear maps that preserve strongly the invertibility of Hilbert space operators
title_full Anti-G-hermiticity preserving linear maps that preserve strongly the invertibility of Hilbert space operators
title_fullStr Anti-G-hermiticity preserving linear maps that preserve strongly the invertibility of Hilbert space operators
title_full_unstemmed Anti-G-hermiticity preserving linear maps that preserve strongly the invertibility of Hilbert space operators
title_sort anti-g-hermiticity preserving linear maps that preserve strongly the invertibility of hilbert space operators
publisher Animo Repository
publishDate 2017
url https://animorepository.dlsu.edu.ph/etd_doctoral/509
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