SUBADDITIVITY OF AN INTEGRAL TRANSFORM FOR POSITIVE OPERATORS IN HILBERT SPACES

Abstract

Abstract For a continuous and positive function $w(\lambda )$ , $\lambda>0$ and $\mu $ a positive measure on $(0,\infty )$ , we consider the integral transform $$ \begin{align*} \mathcal{D}( w,\mu ) ( T) :=\int_{0}^{\infty }w(\lambda) ( \lambda +T) ^{-1}\,d\mu ( \lambda ) , \end{align*} $$ where the integral is assumed to exist for T a positive operator on a complex Hilbert space H. We show among other things that if B, $A>0,$ then $\mathcal {D}( w,\mu ) $ is operator subadditive on $(0,\infty ) $ , that is, $$ \begin{align*} \mathcal{D}( w,\mu ) ( A) +\mathcal{D}( w,\mu) ( B) \geq \mathcal{D}( w,\mu )(A+B). \end{align*} $$ From this, we derive that if $f:[0,\infty )\rightarrow \mathbb {R}$ is an operator monotone function on $[0,\infty )$ , then the function $[ f( t) -f( 0) ] t^{-1}$ is operator subadditive on $( 0,\infty ) .$ Also, if $f:[0,\infty )\rightarrow \mathbb {R}$ is an operator convex function on $[0,\infty )$ , then the function $[ f( t) -f( 0) -f_{+}^{\prime }( 0) t ] t^{-2}$ is operator subadditive on $( 0,\infty ) .$