On the tensor stability of \(s\)-number ideals
type: Item
instance of: scholarly article
title: On the tensor stability of \(s\)-number ideals
zbMATH Open document ID: 0547.47027
DOI: 10.1007/BF01455997
published in: Mathematische Annalen
publication date: 1984-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/163922
review text: Let \(S_{p,q}\) denote the class of operators having approximation numbers belonging to the Lorentz sequence space \(\ell_{p,q}\), \(0<p<\infty\), \(0<q\leq\infty \). We show that these ideals are unstable under suitable tensor norms \(\tau\) for p,q with \(1/q<(1/p)+1/2\), in particular for \(p=q\); i.e. there are \(S_{p,q}\)-operators S,T between Banach spaces such that the tensor product operator \(S{\hat\otimes }_{\tau}T\) does not belong to \(S_{p,q}\), answering a question of Pietsch. These operator ideals \(S_{p,q}\) are, however, ''almost'' stable under arbitrary tensor norms, namely stable up to logarithmic factors. For Hilbert space operators, \(S,T\in S_{p,q}(H,H)\) implies \(S{\hat\otimes }_{\tau}T\in S_{p,q}(H{\hat\otimes }_{\tau}H,H{\hat\otimes }_{\tau}H),\) by contrast, if \(q\leq p\). Using this, factorization theorems, and bilinear interpolation we show that \(S_{p,q}\)-operators are stable under tensor products, if \(1/q\geq (1/p)+1\) and the operators act between dual cotype 2 and cotype 2 Banach spaces. If the cotype conditions are not satisfied, one can still conclude that the Weyl-numbers of the tensor product operator belong to \(\ell_{p,q}\), at least for certain tensor norms.
Mathematics Subject Classification ID: 47L10
Mathematics Subject Classification ID: 47B10
Mathematics Subject Classification ID: 46M05
zbMATH DE Number: 3872116
zbMATH Keywords: tensor stability of s-number ideals
zbMATH Keywords: approximation numbers
zbMATH Keywords: Lorentz sequence space
zbMATH Keywords: tensor norms
zbMATH Keywords: operator ideals
zbMATH Keywords: stable up to logarithmic factors
zbMATH Keywords: factorization theorems
zbMATH Keywords: bilinear interpolation
zbMATH Keywords: dual cotype 2
zbMATH Keywords: Weyl-numbers
author: Hermann König
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3872116