Reverses and variations of Heinz inequality

Abstract

Let $A, B$ be positive definite $n\times n$ matrices. We present several reverse Heinz type inequalities, in particular \begin{align*} \|AX+XB\|_2^2+ 2(\nu-1) \|AX-XB\|_2^2\leq \|A^{\nu}XB^{1-\nu}+A^{1-\nu}XB^{\nu}\|_2^2, \end{align*} where $X$ is an arbitrary $n \times n$ matrix, $\|\cdot\|_2$ is Hilbert-Schmidt norm and $\nu>1$. We also establish a Heinz type inequality involving the Hadamard product of the form \begin{align*} 2|||A^{1\over2}\circ B^{1\over2}|||\leq|||A^{s}\circ B^{1-t}+A^{1-s}\circ B^{t}||| \leq\max\{|||(A+B)\circ I|||,|||(A\circ B)+I|||\}, \end{align*} in which $s, t\in [0,1]$ and $|||\cdot|||$ is a unitarily invariant norm.