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On the Cohen strongly \(p\)-summing multilinear operators. (English) Zbl 1121.47013
Let \(X_{1},\dots,X_{m},Y\) be Banach spaces and \(p\in[1,\infty].\) An \(m\)-linear operator \(T:X_{1}\times\cdots\times X_{m}\rightarrow Y\) is called Cohen strongly \(p\)-summing if there is a constant \(C>0\) such that for any \(x_{1}^{j},\dots,x_{n}^{j}\in X_{j}\) (\(j=1,\dots,m)\) and any \(y_{1}^{\ast},\dots,y_{n}^{\ast}\in Y^{\ast}\), we have that \[ \left\|\left(\langle T(x_{i}^{1},\dots,x_{i}^{m}),y_{i}^{\ast}\rangle\right)_{i}\right\|_{\ell^{1}(n)}\leq C\left(\sum_{i=1}^{n}\prod_{j=1}^{m}\left\| x_{i}^{j}\right\|_{X_{j}}^{p}\right)^{1/p}\sup_{\left\| y\right\|\leq1}\left\| y_{i}^{\ast}(y)\right\|_{\ell^{p^{\ast}}(n)}. \] Using Ky Fan’s lemma, a Pietsch type domination theorem is proved within this class of multilinear operators. Also, a result of Q.-Y. Bu [Contemp. Math. 328, 145–149 (2003; Zbl 1066.47019)] is extended, providing a framework under which every \(p\)-dominated \(m\)-linear operator is Cohen strongly \(p\)-summing.

47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
47H60 Multilinear and polynomial operators
46G25 (Spaces of) multilinear mappings, polynomials
Full Text: DOI
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