In this paper we provide a unifying approach to the study of Banach ideals of linear and multilinear operators defined, or characterized, by the transformation of vector-valued sequences. We investigate and apply the linear and multilinear stabilities of some frequently used classes of vector-valued sequences.

We study the ideals of linear operators between Banach spaces determined by the transformation of vector-valued sequences involving the new sequence space introduced by Karn and Sinha [12] and the classical spaces of absolutely, weakly and Cohen strongly summable sequences. As applications, we prove a new factorization theorem for absolutely summing operators and a contribution to the existence of infinite dimensional spaces formed by non-absolutely summing operators is given.

Considering the successful theory of multiple summing multilinear operators as a prototype, we introduce the classes of multiple Cohen strongly p-summing multilinear operators and polynomials. The adequacy of these classes under the viewpoint of the theory of multilinear and polynomial ideals and holomorphy types is discussed in detail.

It was recently proved by Bayart et al. that the complex polynomial Bohnenblust-Hille inequality is subexponential. We show that, for real scalars, this does no longer hold. Moreover, we show that, if D R,m stands for the real Bohnenblust-Hille constant for m-homogeneous polynomials, then lim sup m D 1/m R,m = 2.2010 Mathematics Subject Classification. 46G25, 47L22, 47H60.

The aim of this paper is to start the study of multilinear generalizations of the classical ideals of linear operators of type p and cotype q. As a first step in a theory we believe will be long and fruitful, we propose a notion of type and cotype of multilinear operators and the resulting classes of such mappings are studied in the setting of the theory of Banach/quasi-Banach ideals of multilinear operators. Distinctions between the linear and the multilinear theories are pointed out, typical multilinear features of the theory are emphasized and many illustrative examples are provided. The classes we introduce are related to the multi-ideals generated by the linear ideals of operators of some type/cotype and are proved to be maximal and Aron-Berner stable.