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Summability of multilinear mappings: Littlewood, Orlicz and beyond. (English) Zbl 1246.46045
In 1930 J. E. Littlewood [Quarterly Journ. (Oxford Series) 1, 164–174 (1930; JFM 56.0335.01)] proved that \[ \left(\sum_{i,j=1}^\infty |A(e_i,e_j)|^{\frac 43}\right)^{\frac 34} \leq \sqrt{2} \| A\| \] for every continuous bilinear form \(A\) on \(c_0 \times c_0\). One year later, H. F. Bohnenblust and E. Hille [Bulletin A. M. S. 37, 37–38, 39 (1931; JFM 57.0276.20)] extended Littlewood’s result to multilinear mappings. In this paper, the authors extend Littlewood’s result to the complex case obtaining the best known constant.
In the next section the coincidence of the space of continuous multilinear forms \({\mathcal L}(E_1,\dots,E_n;\mathbb{K})\) with the space of absolutely \((p; p_1,\dots,p_n)\)-summing multilinear forms \(\Pi_{(p; p_1,\dots,p_n)}(E_1,\dots,E_n;\mathbb{K})\), where \(E_1,\dots,E_n\) are Banach spaces, is studied. It is shown that for \((p; p_1,\dots,p_n) = (1;1,\dots,1,2)\) the spaces are equal if \(E_n'\) has the Littlewood-Orlicz property.
A further theorem shows that \({\mathcal L}(E_1,\dots,E_n;\mathbb{K})=\Pi_{(1;2,\dots,2)}(E_1,\dots,E_n;\mathbb{K})\) if every continuous bilinear form on \(E_i \times E_j\) is 2-dominated for some \(i \not= j\) and \(E_k'\) has the Littlewood-Orlicz property for every \(i\not= k \not= j\). This result is closely related to GT-spaces and spaces of cotype 2.

MSC:
46G25 (Spaces of) multilinear mappings, polynomials
47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
47L22 Ideals of polynomials and of multilinear mappings in operator theory
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