# Operator Lipschitz functions and linear fractional transformations

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.

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Correspondence to A. B. Aleksandrov.

Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 401, 2012, pp. 5–52.

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Aleksandrov, A.B. Operator Lipschitz functions and linear fractional transformations. J Math Sci 194, 603–627 (2013). https://doi.org/10.1007/s10958-013-1550-z

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• DOI: https://doi.org/10.1007/s10958-013-1550-z

### Keywords

• Complex Plane
• Real Line
• Closed Subset
• Lipschitz Function
• Linear Fractional Transformation