Inequalities related to Bourin and Heinz means with a complex parameter

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:426

Article

Bottazzi, T.¹  Elencwajg, R.¹  Larotonda, G.^1,2  Varela, A.^1,2 

[1] Inst Argentino Matemat Alberto P Calderon, RA-1083 Buenos Aires, DF, Argentina

[2] Univ Nacl Gen Sarmiento, Inst Ciencias, Los Polvorines, Buenos Aires, Argentina

关键词: Frobenius norm;    Heinz mean;    Norm inequality;    Complex methods;    Unitarily invariant norm;    Tracial algebra;   

DOI  :  10.1016/j.jmaa.2015.01.046

来源: Elsevier

【 摘 要 】

A conjecture posed by S. Hayajneh and F. Kittaneh claims that given A, B positive matrices, 0 <= t <= 1, and any unitarily invariant norm the following inequality holds vertical bar parallel to A(t)B(1-t) + B(t)A(1-t) vertical bar parallel to <= vertical bar parallel to A(t)B(1-t) + A(1-t)B(t)vertical bar parallel to. Recently, R. Bhatia proved the inequality for the case of the Frobenius norm and for t is an element of[1/4, 3/4]. In this paper, using complex methods we extend this result to complex values of the parameter t = z in the strip {z is an element of C : Re(z) is an element of [1/4, 3/4]}. We give an elementary proof of the fact that equality holds for some z in the strip if and only if A and B commute. We also show a counterexample to the general conjecture by exhibiting a pair of positive matrices such that the claim does not hold for the uniform norm. Finally, we give a counterexample for a related singular value inequality given by s(j)(A(t)B(1-t) + B(t)A(1-t)) <= s(j) (A + B), answering in the negative a question made by K. Audenaert and F. Kittaneh. The methods of proof and examples can be adapted with no modifications to operator algebras (infinite dimensional setting), for instance it follows that the inequality above holds for Hilbertp-Schmidt operators with their Banach algebra norm derived from the infinite trace of B(H). (C) 2015 Elsevier Inc. All rights reserved.

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