Tensorial and Hadamard Product Inequalities for Synchronous Functions

Abstract

Let $H$ be a Hilbert space. In this paper we show among others that, if $f,$ $g$ are synchronous and continuous on $I$ and $A,$ $B$ are selfadjoint with spectra ${Sp}\left( A\right) ,$ ${Sp}\left( B\right) \subset I,$ then% \begin{equation*} \left( f\left( A\right) g\left( A\right) \right) \otimes 1+1\otimes \left( f\left( B\right) g\left( B\right) \right) \geq f\left( A\right) \otimes g\left( B\right) +g\left( A\right) \otimes f\left( B\right) \end{equation*}% and the inequality for Hadamard product% \begin{equation*} \left( f\left( A\right) g\left( A\right) +f\left( B\right) g\left( B\right) \right) \circ 1\geq f\left( A\right) \circ g\left( B\right) +f\left( B\right) \circ g\left( A\right) . \end{equation*}% Let either $p,q\in \left( 0,\infty \right) $ or $p,q\in \left( -\infty ,0\right) $. If $A,$ $B>0,$ then \begin{equation*} A^{p+q}\otimes 1+1\otimes B^{p+q}\geq A^{p}\otimes B^{q}+A^{q}\otimes B^{p}, \end{equation*}% and% \begin{equation*} \left( A^{p+q}+B^{p+q}\right) \circ 1\geq A^{p}\circ B^{q}+A^{q}\circ B^{p}. \end{equation*}