On factorization of Hilbert--Schmidt mappings
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.
scientific article; zbMATH DE number 2157558