Lipschitz type inequalities for noncommutative perspectives of operator monotone functions in Hilbert spaces

Abstract

Assume that $f:[0,\infty )\rightarrow {\mathbb {R}}$ is a continuous function. We can define the perspective $\mathcal {P}_{f}\left( B,A\right)$ by setting

$$\begin{aligned} {\mathcal {P}}_{f}\left( B,A\right) :=A^{1/2}f\left( A^{-1/2}BA^{-1/2}\right) A^{1/2}, \end{aligned}$$

where A,  $B>0.$ We show in this paper among others that

$$\begin{aligned}&\left\| {\mathcal {P}}_{f}\left( B,P\right) -{\mathcal {P}}_{f}\left( A,P\right) \right\| \\&\quad \le \frac{\left\| P\right\| ^{2}\left\| B-A\right\| }{p^{2}} \left\{ \begin{array}{ll} \left( \frac{{\mathcal {P}}_{f}\left( m_{2},p\right) -{\mathcal {P}}_{f}\left( m_{1},p\right) }{m_{2}-m_{1}}\right) &{}\text { if }m_{1}\ne m_{2}, \\ \\ f^{\prime }\left( \frac{m}{p}\right) &{} \text { if }m_{1}=m_{2}=m \end{array} \right. \end{aligned}$$

for all $A\ge m_{1}>0,$ $B\ge m_{2}>0$ and $P\ge p>0$. If f is operator monotone on $[0,\infty)$, then for all $C\ge n_{1}>0,$ $D\ge n_{2}>0,$ $Q>q>0$ we also have

$$\begin{aligned}&\left\| {\mathcal {P}}_{f}\left( Q,D\right) -{\mathcal {P}}_{f}\left( Q,C\right) \right\| \\&\quad \le \frac{\left\| Q\right\| ^{2}\left\| D-C\right\| }{q^{2}} \left\{ \begin{array}{ll} \left[ \frac{{\mathcal {P}}_{f}\left( q,n_{2}\right) -{\mathcal {P}}_{f}\left( q,n_{1}\right) }{n_{2}-n_{1}}\right] &{}\text { if }n_{2}\ne n_{1}, \\ \\ \left[ f\left( \frac{q}{n}\right) -\frac{q}{n}f^{\prime }\left( \frac{q}{n} \right) \right] &{}\text { if }n_{2}=n_{1}=n. \end{array} \right. \end{aligned}$$

Some applications for weighted operator geometric mean and relative operator entropy are also given.