Duality and Lorentz-Marcinkiewicz Operator-Spaces

Abstract

Let S,q be the collection of all compact operators T on a (complex) Hilbert space H such that (INVALID INPUT),q(T) = (P1 n=1((n)sn(T))qn−1)1/q < 1. Here (sn(T)) are the singular numbers of T, 0 < q 1 and :(0,1) ! (0,1) is a continuous function with (1) = 1 and ¯(t):= sups>0((ts)/(s)) < 1 for every t > 0. The special case (t) = t1/p gives the operator space (Sp,q, p,q) introduced in 1967 by H. Triebel [Invent. Math. 4, 275-279 (1967; Zbl 0165.145)]. We characterize the dual of S,q. In particular, we prove that (Sp,q)0 = L(H) for 0 < p < 1 and 0 < q 1, or p = 1 and 0 < q < 1. This complements previous results of C. Merucci [C. R. Acad. Sci., Paris, S´er. A 274, 1163-1166 (1972; Zbl 0238.46037)] and C. Gapaillard and Pham the Lai [Stud. Mat. 49, 129-138 (1974; Zbl 0244.47013)] on duality of Sp,q-spaces.