2014/03/28 by Tamara Bottazzi, Bottazzi, Tamara, Rene Elencwajg +5
Mathematics · #15A42 #15A45 (Primary) #47A30 #47A63 (Secondary) #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA) #math.FA #math.OA #msc:15A42 #msc:15A45 #msc:47A30 #msc:47A63
paper · pdf · doi:10.48550/arxiv.1403.7472
9 pages, 1 figure
arxiv created 2014/03/28 · arxiv updated 2014/03/31
A conjecture posed by S. Hayajneh and F. Kittaneh claims that given A,B positive matrices, 0≤ t≤ 1, and any unitarily invariant norm it holds |||AtB1-t+BtA1-t|||≤|||AtB1-t+A1-tBt|||. Recently, R. Bhatia proved the inequality for the case of the Frobenius norm and for t∈ [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 ∈ \mathbb C: Re(z) ∈ [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 sj(AtB1-t+BtA1-t)≤ sj(A+B), answering in the negative a question made by K. Audenaert and F. Kittaneh.