Some new Jensen’s type inequalities for di¤erentiable log-convex functions of selfadjoint operators in Hilbert spaces under suitable assumptions for the involved operators are given. Applications for particular cases of interest are also provided. 1. Introduction Jensen’s inequality for convex functions is one of the most important result in the Theory of Inequalities due to the fact that many ...

The Barankin bound is generalized to the vector case in the mean square error sense. Necessary and sufficient conditions are obtained to achieve the lower bound. To obtain the result, a simple finite dimensional real vector valued generalization of the Riesz representation theorem for Hilbert spaces is given. The bound has the form of a linear matrix inequality where the covariances of any unbi...

In the first part of paper, we use states on $C^*$-algebras in order to establish some equivalent statements equality triangle inequality, as well parallelogram identity for elements a pre-Hilbert $C^*$-module. We also characterize case inequality adjointable operators Hilbert Then give certain necessary and sufficient conditions Pythagoras two vectors $C^*$-module under assumption that their i...

This paper gives an essentially self-contained exposition (except for appeal to the Lojasiewicz gradient inequality) of geometric invariant theory from a differential viewpoint. Central ingredients are moment-weight inequality (relating Mumford numerical invariants norm moment map), negative flow map squared, and Kempf-Ness function.