Reverse of the grand Furuta inequality and its applications

Abstract

We shall give a norm inequality equivalent to the grand Furuta inequality, and moreover show its reverse as follows: Let $A$ and $B$ be positive operators such that $m \leq B \leq M$ for some positive scalars $m \leq M, m \neq M$ and $h:=\frac{M}{m}$. Then \begin{eqnarray*} \|A^{\frac12}\{A^{-\frac{t}2}(A^{\frac{r}2}B^{\frac{(r-t)\{(p-t)s+r\}}{1-t+r}}A^{\frac{r}2})^{\frac1s}A^{-\frac{t}2}\}^{\frac1p}A^{\frac12}\| \\ \leq K(h^{r-t},\frac{(p-t)s+r}{1-t+r})^{\frac1{ps}} \|A^{\frac{1-t+r}2}B^{r-t}A^{\frac{1-t+r}2}\|^{\frac{(p-t)s+r}{ps(1-t+r)}} \end{eqnarray*} for $0 \leq t \leq 1$, $p \geq 1$, $s \geq 1$ and $r \geq t \geq 0$, where $K(h,p)$ is the generalized Kantorovich constant. As applications, we consider reverses related to the Ando-Hiai inequality.