Multi-ideais simétricos e representação de classes de sequências por ideais de operadores

Abstract

Let X1,...,Xn,X,Y be sequences classes. An n-linear operator A: E1×···×En → F, where E1,...,En,F are Banach spaces, belongs to the ideal of (X1,...,Xn;Y )-summing multilinear operators if (A(x1 j,...,xn j))∞ j=1 ∈ Y(F) whenever (xk j)∞ j=1 ∈ Xk(Ek),k = 1, . . . ,n. In this thesis we compare the ideals of multilinear operators generated by this technique with the ideals generated by other canonical methods, such as the linearization and the factorization methods. We also develop techniques to generate non trivial symmetric ideals of the summing type. The representation of sequence classes by (linear) operator ideals is developed. We say that a sequence class X is ideal-representable if there is an Banach ideal I and a Banach space λ of scalar-valued sequences such that the map u ∈I(λ;E) → (u(ej))∞ j=1 ∈ X(E) is an isometric isomorphism for every Banach space E. We find conditions for X to be ideal-representable and, in this case, we find explicitly an ideal that represents X. Illustrative examples and additional applications of the techniques are provided.