## 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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### References:

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