Concave functions and positive block matrices

초록

Let f (t) be a nonnegative concave function on [0,infinity). For a positive (semidefinite) matrix partitioned into four blocks such that AX = XA or BX = XB, we prove that<br /> <br /> & Vert; f ([A X* X B]) & Vert; <= 2 & Vert; f (A+B/ 2)& Vert;<br /> <br /> for all unitarily invariant norms. The case X=0 is already new, contains two classical trace inequalities due to Rotfel'd and von Neumann, and generalizes an important basic majorization. Our proof is based, and also extends, a theorem of Bourin and Mhanna involving the width of the numerical range of X. For Schatten q-quasinorms, 0 < q < 1, and nonnegative convex functions vanishing at 0, we obtain the reverse inequality.

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