Symmetric ideals of generalized summing multilinear operators

Abstract: Let $X_1, \ldots, X_n,Y$ be classes of Banach spaces-valued sequences. An $n$-linear operator $A$ between Banach spaces belongs to the ideal of $(X_1, \ldots, X_n;Y)$-summing multilinear operators if $(A(x_j^1, \ldots, x_j^n))_{j=1}^\infty$ belongs to $Y$ whenever $(x_j^k)_{j= 1}^\infty$ belongs to $X_k, k = 1, \ldots, n$. In this paper we develop techniques to generate non trivial symmetric ideals of this type. Illustrative examples and additional applications of the techniques are provided.