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*}
Keywords
Convex functions, Hadamard Product, Selfadjoint operators, Tensorial product
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