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|Type:||Artigo de periódico|
|Title:||Mappings between Banach spaces that send mixed summable sequences into absolutely summable sequences|
|Abstract:||In this work we study mappings f from an open subset A of a Banach space E into another Banach space F such that, once a is an element of A is fixed, for mixed (s; q)-summable sequences (x j)(infinity)(j=1) of elements of a fixed neighborhood of 0 in E, the sequence (f (a + x (j)) - f (a))(infinity)(j=1) is absolutely p-summable in F. In this case we say that f is (p; m(s; q))-summing at a. Since for s = q the mixed (s; q)-summable sequences are the weakly absolutely q-summable sequences, the (p; m(q; q))-summing mappings at a are absolutely (p; q)-summing mappings at a. The nonlinear absolutely summing mappings were studied by Matos (see [Math. Nachr. 258 (2003) 71-89]) in a recent paper, where one can also find the historical background for the theory of these mappings. When s = +infinity, the mixed (infinity, q)-summable sequences are the absolutely q-summable sequences. Hence the (p; m(infinity; q))summing mappings at a are the regularly (p; q)-summing mappings at a. These mappings were also studied in [Math. Nachr. 258 (2003) 71-89] and they were important to give a nice characterization of the absolutely (p; q)-summing mappings at a. We show that for q < s < +infinity the space of the (p; m(s; q))-summing mappings at a are different from the spaces of the absolutely (p; q)-summing mappings at a and different from the spaces of regularly (p; q)-summing mappings at a. We prove a version of the Dvoretzky-Rogers theorem for n-homogeneous polynomials that are (p; m(s; q))summing at each point of E. We also show that the sequence of the spaces of such n-homogeneous polynomials, n is an element of N, gives a holomorphy type in the sense of Nachbin. For linear mappings we prove a theorem that gives another characterization of (s; q)-mixing operators in terms of quotients of certain operators ideals. (C) 2004 Elsevier Inc. All rights reserved.|
|Editor:||Academic Press Inc Elsevier Science|
|Appears in Collections:||Artigos e Materiais de Revistas Científicas - Unicamp|
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