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On factorization of Hilbert–Schmidt mappings. (English) Zbl 1066.47062

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.

MSC:

47H60 Multilinear and polynomial operators
47A68 Factorization theory (including Wiener-Hopf and spectral factorizations) of linear operators
46G25 (Spaces of) multilinear mappings, polynomials
46G20 Infinite-dimensional holomorphy
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