Monotonicity for entrywise functions of matrices

Abstract

We characterize real functions $f$ on an interval $(-\alpha ,\alpha )$ for which the entrywise matrix function $[a_{ij}]\mapsto [f(a_{ij})]$ is positive, monotone and convex, respectively, in the positive semidefiniteness order. Fractional power functions are exemplified and related weak majorizations are shown.

23 pages; Section 6 is considerably improved