Operator Lipschitz functions and linear fractional transformations

Abstract

It is known that the function $t^2\sin\frac1t$ is an operator Lipschitz function on the real line ${\mathbb R}$ . We prove that the function sin can be replaced by any operator Lipschitz function f with f(0) = 0. In other words, for every operator Lipschitz function f, the function $t^2 f(\frac1t)$ is also operator Lipschitz if f(0) = 0. The function f can be defined on an arbitrary closed subset of the complex plane ${\mathbb C}$ . Moreover, the linear fractional transformation $\frac1t$ can be replaced by every linear fractional transformation ϕ. In this case, we assert that the function $\dfrac{f\circ\varphi}{\varphi^{\,\prime}}$ is operator Lipschitz for every operator Lipschitz function f provided that f(ϕ( ∞ )) = 0. Bibliography: 12 titles.