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Mixing multilinear operators. (English) Zbl 1301.46021

In his monograph [Operator ideals. Berlin: VEB Deutscher Verlag der Wissenschaften (1978; Zbl 0399.47039)], A. Pietsch introduced the class of all \((q,p)\)-mixing linear operators, here denoted by \(\mathcal{M}_{(q,p)}\), and proved some inclusions and composition results for this class. In this paper, the author extends this concept for a multilinear scenario, introducing the class of all \(((q_1,p_1),\dots, (q_n,p_n))\)-mixing \(n\)-linear operators, denoted by \(\mathcal{M}^n_{(q_1,p_1),\dots, (q_n,p_n)}\).
As in Pietsch’s work, the author also shows some inclusion and composition results for this new class. Among them he proves that, under the expected conditions on the parameters, \[ \mathcal{L}\left(\mathcal{M}_{(q_1,p_1)},\dots, \mathcal{M}_{(q_n,p_n)} \right) \subset \mathcal{M}^n_{(q_1,p_1),\dots, (q_n,p_n)}\;\text{ and} \]
\[ \|\cdot\|_{\mathcal{M}^n_{(q_1,p_1),\dots, (q_n,p_n)}} \leq \|\cdot\|_{\mathcal{L}\left(\mathcal{M}_{(q_1,p_1)},\dots, \mathcal{M}_{(q_n,p_n)}\right)}, \] where \(\mathcal{L}\left(\mathcal{M}_{(q_1,p_1)},\dots, \mathcal{M}_{(q_n,p_n)} \right)\) is the multi-ideal constructed by factorization method from \(\mathcal{M}_{(q_1,p_1)},\dots, \mathcal{M}_{(q_n,p_n)}\) (see [G. Botelho, Note Mat. 25(2005/2006), No. 1, 69–102 (2006; Zbl 1223.46047)], p. 75).

MSC:

46G25 (Spaces of) multilinear mappings, polynomials
47H60 Multilinear and polynomial operators
47L22 Ideals of polynomials and of multilinear mappings in operator theory
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References:

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