Perspectives of matrix convex functions

Abstract

In this paper, we generalize the main results of [Effros EG, (2009) Proc Natl Acad. Sci USA 106:1006–1008]. Namely, we provide the necessary and sufficient conditions for jointly convexity of perspective functions and generalized perspective functions.

f-Divergence functional of operator log-convex functions

“…[9,10] A more general version of g, the non-commutative f -divergence functional was defined in Moslehian and Kian [11] to be…”

“…(2) f is operator log-convex; If f is operator convex, then (2) defines [9,10] the perspective of f denoted by g, i.e.…”

Non-commutative perspectives

“…al. [3] noticed that virtually the same proof applies without any commutativity conditions. We include the proof for the convenience of the reader.…”

A Perspective Approach for Characterization of Lieb Concavity Theorem

“…In particular, by using the affine version of Hansen-Pedersen-Jensen inequality [H,Theorem 2.1] we could prove the following theorem (see [ENE,Theorem 2.2…”

“…We proved in [ENE] the necessary and sufficient conditions for joint convexity of a fully noncommutative generalized perspective function. For the next applications, we now refine [ENE,Theorem 2.5…”