Inequalities for quadratic operator perspective of convex functions and bounded linear operators on Hilbert spaces

Abstract

In this paper we introduce the concept of quadratic operator perspective for a continuous function $\text{Φ}$ defined on the positive semi-axis of real numbers, the invertible operator $T$ and operator $V$ on a Hilbert space by

$${◯}_{\Phi }(V,T):=T^{*}\Phi (|V{T}^{-1}{|}^2)T.$$

This generalize the quadratic weighted operator geometric mean of $(T,V)$ defined by

$$T{\odot }_{v}V:=||V{T}^{-1}{|}^{v}T{|}^2$$

for $v\in [0,1]$ and the quadratic relative operator entropy defined by

$$\odot \left(T\text{|}V\right):=T^{*}\ln \left({\left|V{T}^{-1}\right|}^2\right)T.$$

Some inequalities for this perspective of convex functions are established. Applications for quadratic weighted operator geometric mean and quadratic relative operator entropy are also provided.