Multilinear Operator Ideals Associated to Saphar Type (n + 1)-tensor Norms

Abstract

We study an (n + 1)-tensor norm gr,s generalizing Saphar's classic norm to (n+1)-fold tensor products. We characterize the maps in the minimal and maximal multilinear operator ideals related to gr,s in the sense of Defant and Floret.Mathematics Subject Classification. Primary 46M05; 46A32. Keywords. (n + 1)-fold tensor products, gr,s-nuclear and gr,s-integral n-linear operators, t-absolutely summing n-linear operators, Ultraproducts.In [11] Pietsch proposed building a systematic theory of ideals of multilinear mappings between Banach spaces, similar to the already well-developed one regarding linear maps, as a first step to study ideals of more general non linear operators.A systematic development of this program has not been initiated until the works [4] and [5] of Floret, mainly motivated by the potential applications of the new theory to infinite holomorphy. Thus, classic notions of maximal operator ideals with their associated α-tensor norm, dual tensor norm α and the related α-nuclear and α-integral operators can be extended to the framework of multilinear operator ideals and multiple tensor products.However, as far as we know, only a few concrete examples of multi-tensor norms have been considered to check and apply the general concepts of the extended theory beyond the injective and projective ones or the introduction of generalized α-nuclear operators (for example, the previous work in [9] stops at this point). In this sense we have studied in [8] an (n + 1)-fold tensor norm generalizing both Lapresté's classic norm as well as Matos's multi-tensor norm defined in [9].The purpose of this paper is to study an (n + 1)-fold tensor norm naturally extending the well-known tensor norm g p of Saphar. This new tensor norm on tensor products n+1 j=1 E j , 1 ≤ n ∈ N, of n + 1 normed spaces E j , will depend on two vector parameters r := (r 1 , r 2 , · · · , r k ) and s := (r k+1 , r k+2 , · · · , r n+1 ) and will be Partially supported by the MEC and FEDER project MTM2004-02262 and AVCIT group 03/050.