E-Operator Ideals Determined by Banach Spaces with Unconditional Bases

Abstract

Let E be a Banach space with an 1-unconditional basis. In this paper, we introduce the notions of E-summing operators and E-dominated operators. We show that the ideals of E-summing operators and E-dominated operators are maximal Banach operator ideals. Characterizations of the ideals of E-summing operators and E-dominated operators are given in terms of the ideals of E-nuclear operators and uniformly E-nuclear operators, respectively. The finite E-compact and finite uniformly E-compact norms are also investigated. We show that the finite E-compact (respectively, finite uniformly E-compact) and the E-compact (respectively, uniformly E-compact) quasi-norms on finite rank operators are equivalent.