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

• Published in: Linear algebra and its applications Vol. 430; no. 11; pp. 2961 - 2974
• Main Author: JOCIC, Danko R
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier Inc 01.06.2009; Elsevier
• Subjects: [formula omitted] invariant norms for spaces of operator valued functions; Algebra; Exact sciences and technology; Interpolation norms; Linear and multilinear algebra, matrix theory; Mathematical analysis; Mathematics; Operator theory; Schatten ideals; Sciences and techniques of general use; Transformers norms and spaces; 47B10; 46E40; 47D50; 47B20; Interpolation norms; Schatten ideals; 47A63; 47A30; invariant norms for spaces of operator valued functions; 46B70; 47B49; 47B47; Transformers norms and spaces; Invariant; valued functions; Ideal; Bounded operator; Normed space; Matrix inequality; Invariant norms for spaces of operator; Matrix function; Linear operator; Hilbert space
• ISSN: 0024-3795
• DOI: 10.1016/j.laa.2009.01.011

For the class of transformers acting as X → ∫ Ω A t X B t d μ ( t ) on the space of bounded Hilbert space operators we give formulae for its norm on the Hilbert–Schmidt class (1) X → ∫ Ω A t X B t d μ ( t ) B ( C 2 ( H ) ) = lim n → ∞ ∫ Ω 2 n tr ∏ k = 1 n A t n + 1 - k ∗ A s n + 1 - k tr ∏ k = 1 n B s k B t k ∗ ∏ k = 1 n d μ ( s k ) d μ ( t k ) 2 n , whenever ∫ Ω ‖ A t ‖ p ‖ B t ‖ 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 = ) 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.