Refinements and Reverses of Tensorial and Hadamard Product Inequalities for Selfadjoint Operators in Hilbert Spaces Related to Young's Result
Abstract
Let $H$ be a Hilbert space. In this paper we show among others that, if the
selfadjoint operators $A$ and $B$ satisfy the condition $0$ $<$ $m\leq A,$ $B\leq
M,$ for some constants $m,$ $M,$ then
\begin{align*}
0& \leq \frac{m}{M^{2}}\nu \left( 1-\nu \right) \left( \frac{A^{2}\otimes
1+1\otimes B^{2}}{2}-A\otimes B\right) \\
& \leq \left( 1-\nu \right) A\otimes 1+\nu 1\otimes B-A^{1-\nu }\otimes
B^{\nu } \\
& \leq \frac{M}{m^{2}}\nu \left( 1-\nu \right) \left( \frac{A^{2}\otimes
1+1\otimes B^{2}}{2}-A\otimes B\right)
\end{align*}
for all $\nu \in \left[ 0,1\right] .$ We also have the inequalities for
Hadamard product
\begin{align*}
0& \leq \frac{m}{M^{2}}\nu \left( 1-\nu \right) \left( \frac{A^{2}+B^{2}}{2}%
\circ 1-A\circ B\right) \\
& \leq \left[ \left( 1-\nu \right) A+\nu B\right] \circ 1-A^{1-\nu }\circ
B^{\nu } \\
& \leq \frac{M}{m^{2}}\nu \left( 1-\nu \right) \left( \frac{A^{2}+B^{2}}{2}%
\circ 1-A\circ B\right)
\end{align*}
for all $\nu \in \left[ 0,1\right] .$
Keywords
Tensorial product, Hadamard Product, Selfadjoint operators, Convex functions
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