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

AbstractIn 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-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