On the compactness of weighted shift operators
type: Item
instance of: scholarly article
title: On the compactness of weighted shift operators
zbMATH Open document ID: 0945.47020
author: Malika Aboufatima
author: Mohamed Houimdi
published in: Acta Mathematica Vietnamica
publication date: 1999-09-14T00:00:00Z
review text: This note presents a proof of a rather simple folklore result: Let \(H\) be a Hilbert space and \((A_n)_n\) a sequence of operators \(H \to H\); let \(H^\infty\) denote the ``Hilbertian direct sum'' of a countable quantity of copies of \(H\). Then the operator \(A: H^\infty \to H^\infty\) given by \(A((x_n)_n)=(A_n x_n)_n\) is compact if and only if each \(A_n\) is compact and \(\lim_{n \to \infty} \|A_n \|=0\). Unfortunately, there are even some obscure points in this as presented: at some point (proof of lemma 2.3) it is asserted that \(H^\infty\) coincides with the projective tensor product \(H \otimes_\pi H\), which is wrong if \(H^\infty\) means, as seems to follow from the equality at the introduction, the Hilbert space \(\ell_2(H)\).
Mathematics Subject Classification ID: 47B37
Mathematics Subject Classification ID: 47B07
zbMATH DE Number: 1310096
zbMATH Keywords: compactness
zbMATH Keywords: weighted shift operators
zbMATH Keywords: Hilbertian direct sum
zbMATH Keywords: projective tensor product
reviewed by: Jesús M. F. Castillo
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1310096