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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.

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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