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On factorization of Schatten class type mappings. (English) Zbl 1113.46040

Let \(H_1 ,\dots , H_n\) be Hilbert spaces and \(F\) a Banach space. The author shows that the space \(L(S_2 )(H_1 ,\dots ,H_n ;F)\) of continuous multilinear mappings of Schatten class type \({\mathcal S}_2\) coincides with the space \(L_{d,p}(H_1, \dots ,H_n;F)\) of continuous multilinear \(p\)-dominated mappings, if \(1 \leq p \leq 2\). (See H.–A.Braunß and H.Junek [Note Mat.10, No.1, 47–58 (1990; Zbl 0773.46020)] and R.Alencar and M.C.Matos, “Some classes of multilinear mappings between Banach spaces” (Publicaciones del Departamento de Análisis Matemático 12, Universidad Complutense de Madrid, Madrid) (1989)] for definitions.) Moreover, an analogous result is true for spaces of \(m\)-homogeneous polynomials of these types.
The main results of this article are characterizations of multilinear mappings and polynomials of type \({\mathcal S}_2\) using factorizations through \(L_1\) and \(L_{\infty}\) spaces. The results are applied to obtain a factorization result for holomorphic mappings of Schatten class type \({\mathcal S}_2\).

MSC:

46G25 (Spaces of) multilinear mappings, polynomials
46G20 Infinite-dimensional holomorphy
47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
46T25 Holomorphic maps in nonlinear functional analysis

Citations:

Zbl 0773.46020
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References:

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