Interpolation norms between row and column spaces and the norm problem for elementary operators

Abstract

For the class of transformers acting as X → ∫Ω At X Bt d μ (t) on the space of bounded Hilbert space operators we give formulae for its norm on the Hilbert-Schmidt class(1)(fenced(X → ∫Ω At X Bt d μ (t))B (C2 (H)); = under(lim, n → ∞) root(∫Ω2 n tr fenced(underover(∏, k = 1, n) Atn + 1 - k* Asn + 1 - k) tr fenced(underover(∏, k = 1, n) Bsk Btk*) underover(∏, k = 1, n) d μ (sk) d μ (tk), 2 n),)whenever ∫Ω {norm of matrix} At {norm of matrix}p {norm of matrix} Bt {norm of matrix}p d μ (t) < ∞ for some p > 0. We also estimate from below its norm on the other Schatten classes. This answers a question of characterizing (θ =) frac(1, 2) interpolation norm between column and row space norm for operator valued functions, with the discrete case providing the solution of the norm problem for elementary operators acting on the Hilbert-Schmidt class. © 2009 Elsevier Inc. All rights reserved.