Mappings between Banach spaces that send mixed summable sequences into absolutely summable sequences

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 (xj )∞j =1 of elements of a fixed neighborhood of 0 in E, the sequence (f (a + xj ) − 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-homogeneouspolynomials, 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.  2004 Elsevier Inc. All rights reserved