On factorization of Hilbert--Schmidt mappings
type: Item
instance of: scholarly article
title: On factorization of Hilbert--Schmidt mappings
zbMATH Open document ID: 1066.47062
author: Cristiane A. Mendes
published in: Journal of Mathematical Analysis and Applications
publication date: 2005-04-18T00:00:00Z
review text: Let \(H_1, \ldots, H_n\), and \(G\) be Hilbert spaces. A continuous multilinear mapping \(T \in L(H_1, \ldots, H_n;G)\) is of Hilbert-Schmidt type if for each \(k=1, \ldots , n\) there exists an orthonormal basis \((h^{(k)}_j)\) of \(H_k\) such that \[ \| T \| _{HS}^2 := \sum_{j_1,\ldots,j_n} \| T(h^{(1)}_{j_1}),\ldots, h^{(n)}_{j_n})\| ^2 < \infty. \] It is easy to see that this number is independent from the choices of the orthonormal bases and this norm corresponds to an inner product. Here are the main results. Theorem. Let \(T \in L(H_1, \dots, H_n;G)\) be given. If there are \({\mathcal L}_\infty \) spaces \(X_j\), operators \(S_j \in L(H_j;X_j)\), and an \(R \in L(X_1, \dots, X_n;G)\) such that \(T = R \circ (S_1, \dots , S_n)\), then \(T\) is a mapping of Hilbert-Schmidt type. Theorem. Let \(T \in L(H_1, \dots, H_n;G)\) a mapping of Hilbert-Schmidt type. For every infinite-dimensional Banach space \(Z\) there are \(S \in L(H_1, \dots, H_n;Z)\) and \(V \in L(Z;G)\) compact and 2-summing such that \(T=V\circ S\). In the last section, the author shows that for every holomorphic mapping \(f\) of Hilbert-Schmidt type (in the sense of Nachbin) defined on an open subset \(U \subseteq H\) and \(h_0 \in U\), there are an open subset \(U_0\), an \({\mathcal L}_\infty\) space \(X\), a holomorphic mapping \(g \in H(U_0;X)\), and an operator \(V \in L(X;G)\) such that \(h_0 \in U_0 \subseteq U\) and \(f = V\circ g\) on \(U_0\). Moreover, for holomorphic Hilbert-Schmidt mappings of bounded type there are global factorizations.
Mathematics Subject Classification ID: 47H60
Mathematics Subject Classification ID: 47A68
Mathematics Subject Classification ID: 46G25
Mathematics Subject Classification ID: 46G20
zbMATH DE Number: 2157558
zbMATH Keywords: Hilbert-Schmidt mappings
zbMATH Keywords: homogeneous polynomials
zbMATH Keywords: holomorphic mappings
reviewed by: Hans-Andreas Braunß
MaRDI profile type: Publication
full work available at URL: j.jmaa.2004.09.008
OpenAlex ID: W2076111337
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DOI: 10.1016/J.JMAA.2004.09.008
description: scientific article; zbMATH DE number 2157558