On certain basis connected with operator and its applications
type: Item
instance of: scholarly article
title: On certain basis connected with operator and its applications
zbMATH Open document ID: 1320.47002
author: Paweł Wójcik
published in: Journal of Mathematical Analysis and Applications
publication date: 2014-11-19T00:00:00Z
review text: Let \(B(H,K)\) denote the Banach space of all bounded linear operators between two complex Hilbert spaces \(H\) and \(K\) and let \({\mathcal S}(H)\) be the unit sphere of \(H\). Given an operator \(T\in B(H,K)\), a set \(\{x_k\in {\mathcal S}(H):k\in{\mathcal K}\}\) is called a \(T\)-orthonormal set if \[ x_k\perp x_j\text{ and }Tx_k\perp Tx_j,~k,j\in{\mathcal K},~k\not=j, \] and is called a \(T\)-orthonormal basis if it is a \(T\)-orthonormal set and its linear span is dense in \(H\). The author investigates the existence of such a basis and establishes some permanence properties of it. He then shows that such a basis plays an important role in describing the approximately orthogonality preserving operators, and gives some applications of his results. Furthermore, he answers the question whether a linear operator which approximately preserves orthogonality must be close to an orthogonality preserving one.
reviewed by: Abdellatif Bourhim
Mathematics Subject Classification ID: 47A05
zbMATH DE Number: 6370940
zbMATH Keywords: operators on Hilbert spaces
zbMATH Keywords: approximately orthogonality preserving operators
zbMATH Keywords: stability
MaRDI profile type: Publication
full work available at URL: j.jmaa.2014.10.075
OpenAlex ID: W2020069449
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DOI: 10.1016/J.JMAA.2014.10.075
description: scientific article; zbMATH DE number 6370940