Superadditivity of convex integral transform for positive operators in Hilbert spaces

Abstract

For a continuous and positive function $w\left( \lambda \right) ,$ $\lambda >0$ and $\mu$ a positive measure on $(0,\infty )$ we consider the following convex integral transform

$$\begin{aligned} \mathcal {C}\left( w,\mu \right) \left( T\right) :=\int _{0}^{\infty }w\left( \lambda \right) T^{2}\left( \lambda +T\right) ^{-1}d\mu \left( \lambda \right) , \end{aligned}$$

where the integral is assumed to exist for T a positive operator on a complex Hilbert space H. We show among other that, for all A,  $B>0$ with $BA+AB\ge 0,$

$$\begin{aligned} \mathcal {C}(w,\mu )\left( A+B\right) \ge \mathcal {C}(w,\mu )\left( A\right) +\mathcal {C}(w,\mu )\left( B\right) . \end{aligned}$$

In particular, we have for $r\in (0,1],$ the power inequality

$$\begin{aligned} \left( A+B\right) ^{r+1}\ge A^{r+1}+B^{r+1} \end{aligned}$$

and the logarithmic inequality

$$\begin{aligned} \left( A+B\right) \ln \left( A+B\right) \ge A\ln A+B\ln B. \end{aligned}$$

Some examples for operator monotone and operator convex functions and integral transforms $\mathcal {C}\left( \cdot ,\cdot \right)$ related to the exponential and logarithmic functions are also provided.