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Subadditivity of eigenvalue sums. (English) Zbl 1089.47010

Summary: Let \(f(t)\) be a nonnegative concave function on \(0 \leq t <\infty\) with \(f(0)=0\), and let \(X, Y\) be \(n\times n\) matrices. Then it is known that \(\| f(| X+Y|)\|_1\leq \| f(| X|)\|_1 +\| f(| Y|)\|_1\), where \(\| \cdot \|_1\) is the trace norm. We extend this result to all unitarily invariant norms and prove some inequalities of eigenvalue sums.

MSC:

47A30 Norms (inequalities, more than one norm, etc.) of linear operators
15A42 Inequalities involving eigenvalues and eigenvectors
47A75 Eigenvalue problems for linear operators
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