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A norm inequality for pairs of commuting positive semidefinite matrices. (English) Zbl 1326.15030

Summary: For \(i=1,\dots,k\), 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, \[ \left|\left\|\sum\limits_{i=1}^kA_iB_i\right|\right\|\leq\left|\left\|\left(\sum\limits_{i=1}^kA_i\right)\,\left(\sum\limits_{i=1}^kB_i\right)\right|\right\|. \] The \(k=2\) case was recently conjectured by S. Hayajneh and F. Kittaneh [Bull. Aust. Math. Soc. 88, No. 3, 384–389 (2013; Zbl 1305.15021)] and proven by them for the trace norm and the Hilbert-Schmidt norm. A simple application of this norm inequality answers a question of J.-C. Bourin [Int. J. Math. 20, No. 6, 679–691 (2009; Zbl 1181.15030)] in the affirmative.

MSC:

15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
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