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Journal of Mathematical Sciences

, Volume 194, Issue 6, pp 603–627 | Cite as

Operator Lipschitz functions and linear fractional transformations

  • A. B. Aleksandrov
Article
  • 43 Downloads

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.

Keywords

Complex Plane Real Line Closed Subset Lipschitz Function Linear Fractional Transformation 
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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.St. Petersburg Department of the Steklov Mathematical InstituteSt.PetersburgRussia

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