Geodesic
Some remarks concerning operator Lipschitz functions
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 47, Tome 480 (2019), pp. 26-47 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

We consider examples of operator Lipschitz functions $f$ for which the operator Lipschitz seminorm $\|f\|_{\mathrm{OL}(\mathbb{R})}$ coincides with the Lipschitz seminorm $\|f\|_{\mathrm{Lip}(\mathbb{R})}$. In particular, we consider the operator Lipschitz functions such that $f'(0)=\|f\|_{\mathrm{OL}(\mathbb{R})}$. It is well known that every function $f$ whose the derivative $f'$ is positive definite has this property. In the paper it is proved that there are other functions having this property. It is also shown that the identity $|f'(t_0)|=\|f\|_{\mathrm{OL}(\mathbb{R})}$ implies that the derivative of $f$ is continuous at $t_0$. In fact, a more general statement is established concerning commutator Lipschitz functions on a closed subset of the complex plane. 
@article{ZNSL_2019_480_a1,
     author = {A. B. Aleksandrov},
     title = {Some remarks concerning operator {Lipschitz} functions},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {26--47},
     year = {2019},
     volume = {480},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2019_480_a1/}
}
TY  - JOUR
AU  - A. B. Aleksandrov
TI  - Some remarks concerning operator Lipschitz functions
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2019
SP  - 26
EP  - 47
VL  - 480
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2019_480_a1/
LA  - ru
ID  - ZNSL_2019_480_a1
ER  - 
%0 Journal Article
%A A. B. Aleksandrov
%T Some remarks concerning operator Lipschitz functions
%J Zapiski Nauchnykh Seminarov POMI
%D 2019
%P 26-47
%V 480
%U http://geodesic.mathdoc.fr/item/ZNSL_2019_480_a1/
%G ru
%F ZNSL_2019_480_a1
A. B. Aleksandrov. Some remarks concerning operator Lipschitz functions. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 47, Tome 480 (2019), pp. 26-47. http://geodesic.mathdoc.fr/item/ZNSL_2019_480_a1/

[1] A. B. Aleksandrov, “Operatorno lipshitsevy funktsii i modelnye prostranstva”, Zap. nauchn. sem. POMI, 416, 2013, 5–58

[2] A. B. Aleksandrov, “Kommutatorno lipshitsevy funktsii i analiticheskoe prodolzhenie”, Zap. nauchn. sem. POMI, 434, 2015, 5–18

[3] A. B. Aleksandrov, V. V. Peller, “Operatorno lipshitsevy funktsii”, UMN, 71:4(430) (2016), 3–106 | DOI | MR | Zbl

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

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

[6] 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

[7] R. Felps, Lektsii o teoremakh Shoke, Mir, M., 1968

[8] G. Pisier, Similarity problems and completely bounded maps, Includes the solution to “The Halmos problem”, Lecture Notes in Mathematics, 1618, Second, expanded edition, Springer-Verlag, Berlin, 2001 | DOI | MR | Zbl