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.