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 ∈ A is fixed, for mixed ( s ; q ) -summable sequences ( x j ) j = 1 ∞ of elements of a fixed neighborhood of 0 in E, the sequence ( f ( a + x j ) − f ( a ) ) 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 = + ∞ , the mixed ( ∞ , q ) -summable sequences are the absolutely q-summable sequences. Hence the ( p ; m ( ∞ ; 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 < + ∞ 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 ∈ 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.