TY - JOUR

T1 - Some Remarks Concerning Operator Lipschitz Functions

AU - Александров, Алексей Борисович

N1 - Publisher Copyright: © 2020, Springer Science+Business Media, LLC, part of Springer Nature.

PY - 2020/11/1

Y1 - 2020/11/1

N2 - We consider examples of operator Lipschitz functions f for which the operator Lipschitz seminorm ‖f‖OL(ℝ) coincides with the Lipschitz seminorm ‖f‖Lip(ℝ). In particular, we consider the operator Lipschitz functions f such that |f ' (0)| = ‖f‖OL(ℝ). It is well known that f has this property if its derivative f′ is positive definite. It is proved in this paper that there are other functions having this property. It is also shown that the identity |f ' (t0)| = ‖f‖OL(ℝ) implies the continuity of f at t0. In fact, a more general statement is established concerning commutator Lipschitz functions on a closed subset of the complex plane.

AB - We consider examples of operator Lipschitz functions f for which the operator Lipschitz seminorm ‖f‖OL(ℝ) coincides with the Lipschitz seminorm ‖f‖Lip(ℝ). In particular, we consider the operator Lipschitz functions f such that |f ' (0)| = ‖f‖OL(ℝ). It is well known that f has this property if its derivative f′ is positive definite. It is proved in this paper that there are other functions having this property. It is also shown that the identity |f ' (t0)| = ‖f‖OL(ℝ) implies the continuity of f at t0. In fact, a more general statement is established concerning commutator Lipschitz functions on a closed subset of the complex plane.

UR - http://www.scopus.com/inward/record.url?scp=85093826058&partnerID=8YFLogxK

U2 - 10.1007/s10958-020-05078-4

DO - 10.1007/s10958-020-05078-4

M3 - Article

AN - SCOPUS:85093826058

VL - 251

SP - 176

EP - 189

JO - Journal of Mathematical Sciences

JF - Journal of Mathematical Sciences

SN - 1072-3374

IS - 2

ER -