The Relations between Volume Ratios and New Concepts of GL Constants

In this paper we investigate a property named GL(p, q) which is closely related to the Gordon-Lewis property. Our results on GL(p, q) are then used to estimate volume ratios relative to lp, 1 < p ≤ ∞, of unconditional direct sums of Banach spaces. Introduction In this paper we investigate a property named GL(p, q), 1 ≤ p ≤ ∞, 1 ≤ q ≤ ∞, closely related to the Gordon-Lewis property GL, and the behavior of p-summing norms of operators defined on direct sums of Banach spaces in the sense of an unconditional basis. These results are then used to estimate the volume ratios vr(X, lp), 1 < p ≤ ∞, where X is a finite direct sum of finite dimensional spaces. A Banach space is said to have GL(p, q), 1 ≤ p ≤ ∞, 1 ≤ q ≤ ∞, if there is a constant C so that iq(T ) ≤ Cπp(T ∗) for every finite rank operator T from an arbitrary Banach space to X. Here πp denotes the p-summing norm and iq the q-integral norm. This property was also considered by Reisner [27], note however the slight difference in the notation: Our GL(p, q) corresponds to his q′, p′-GL-space. We now wish to discuss the arrangement and contents of this paper in greater detail. In Section 1 of the paper we investigate the basic properties of GL(p, q) and prove some inequalities for p-summing operators, respectively q-integral operators, defined on, Supported in part by the fund for the promotion of research in the Technion and by the VPR fund. Supported in part by the Danish Natural Science Research Council, grants 9503296 and 9600673. 1 respectively with range in, a direct sum of Banach spaces in the sense of an unconditional basis. These inequalities are then used to prove that if (Xn) is a sequence of Banach spaces with uniformly bounded GL(p, q)-constants and X is the direct sum of the Xn’s in the sense of a p-convex and q-concave unconditional basis, then X has GL(p, q) as well. More generally we obtain that if Y is a Banach space with GL(p, q) and L is a p-convex and q-concave Banach lattice, then L(Y ) has GL(p, q). K(L) and Kq(L) denote the p-convexity and q-concavity constants of L respectively. In Section 2 we combine the results of Section 1 with those of [6] to obtain some estimates of volume ratios. One of our results, Theorem 2.5, has the following geometric consequence: Let L be a p-convex and q-concave Banach lattice having an n-dimensional Banach space Y = (R, ‖ · ‖) as an isometric quotient. Let 1 ≤ p, q ≤ ∞, 1 p + 1 p = 1, then there are ndimensional linear quotients Vp and Vq of Blp and Blq respectively, so that Vq ⊆ BY ⊆ Vp for which ( |Vp| |Vq| ) 1 n ≤ c √ p′ glp,q(L) ≤ c √ p′ K(L)Kq(L). If X is a finite direct sum of nk-dimensional Banach spaces Xk, 1 ≤ k ≤ m in the sense of a finite 1-unconditional basis, then we prove that