A NORM INEQUALITY FOR PAIRS OF COMMUTING POSITIVE SEMIDEFINITE MATRICES

For i = 1, ... , k, let A(i) and B-i be positive semidefinite matrices such that, for each i, A(i) commutes with B-i. It is shown that, for any unitarily invariant norm, vertical bar vertical bar vertical bar Sigma(k)(i-1)A(i)B(i)vertical bar vertical bar vertical bar <= vertical bar vertical bar vertical bar (Sigma(k)(i-1)A(i)(Sigma B-k(i-1)i)vertical bar vertical bar vertical bar. The k = 2 case was recently conjectured by Hayajneh and Kittaneh and proven by them for the trace norm and the Hilbert-Schmidt norm. A simple application of this norm inequality answers a question of Bourin in the affirmative.