Geodesic
Operator Lipschitz functions and linear fractional transformations
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 40, Tome 401 (2012), pp. 5-52 
A. B. Aleksandrov. Operator Lipschitz functions and linear fractional transformations. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 40, Tome 401 (2012), pp. 5-52. http://geodesic.mathdoc.fr/item/ZNSL_2012_401_a0/
@article{ZNSL_2012_401_a0,
     author = {A. B. Aleksandrov},
     title = {Operator {Lipschitz} functions and linear fractional transformations},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {5--52},
     year = {2012},
     volume = {401},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2012_401_a0/}
}
TY  - JOUR
AU  - A. B. Aleksandrov
TI  - Operator Lipschitz functions and linear fractional transformations
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2012
SP  - 5
EP  - 52
VL  - 401
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2012_401_a0/
LA  - ru
ID  - ZNSL_2012_401_a0
ER  - 
%0 Journal Article
%A A. B. Aleksandrov
%T Operator Lipschitz functions and linear fractional transformations
%J Zapiski Nauchnykh Seminarov POMI
%D 2012
%P 5-52
%V 401
%U http://geodesic.mathdoc.fr/item/ZNSL_2012_401_a0/
%G ru
%F ZNSL_2012_401_a0

Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

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 $\varphi$. In this case, we assert that the function $\dfrac{f\circ\varphi}{\varphi'}$ is operator Lipschitz for every operator Lipschitz function $f$ provided $f(\varphi(\infty))=0$.

[1] A. B. Aleksandrov, V. V. Peller, “Functions of perturbed unbounded self-adjoint operators. Operator Bernstein type inequalities”, Indiana Univ. Math. J., 59:4 (2010), 1451–1490 | DOI | MR | Zbl

[2] A. B. Aleksandrov, V. V. Peller, “Estimates of operator moduli of continuity”, J. Funct. Anal., 261:10 (2011), 2741–2796 | DOI | MR | Zbl

[3] A. B. Aleksandrov, V. V. Peller, “Operator and commutator moduli of continuity for normal operators”, Proc. London Math. Soc. (to appear)

[4] K. Iosida, Funktsionalnyi analiz, Mir, M., 1967 | MR

[5] B. E. Johnson, J. P. Williams, “The range of a normal derivation”, Pacific J. Math., 58 (1975), 105–122 | DOI | MR | Zbl

[6] H. Kamowitz, “On operators whose spectrum lies on a circle or a line”, Pacific J. Math., 20 (1967), 65–68 | DOI | MR | Zbl

[7] E. Kissin, V. S. Shulman, “On a problem of J. P. Williams”, Proc. Amer. Math. Soc., 130 (2002), 3605–3608 | DOI | MR | Zbl

[8] E. Kissin, V. S. Shulman, “Classes of operator-smooth functions. I. Operator-Lipschitz functions”, Proc. Edinb. Math. Soc. (2), 48 (2005), 151–173 | DOI | MR | Zbl

[9] E. Kissin, V. S. Shulman, “On fully operator Lipschitz functions”, J. Funct. Anal., 253 (2007), 711–728 | DOI | MR | Zbl

[10] E. Kissin, V. S. Shulman, L. B. Turowska, “Extension of operator Lipschitz and commutator bounded functions”, The extended field of operator theory, Oper. Theory Adv. Appl., 171, Birkhäuser, Basel, 2007, 225–244 | DOI | MR

[11] I. M. Stein, Singulyarnye integraly i differentsialnye svoistva funktsii, Mir, M., 1973 | MR | Zbl

[12] J. P. Williams, “Derivation ranges: open problems”, Topics in Modern Operator Theory (Timisoara/Herculane, 1980), Operator Theory Adv. Appl., 2, Birkhäuser, Basel–Boston, MA, 1981, 319–328 | DOI | MR