Some Remarks Concerning Operator Lipschitz Functions
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SOME REMARKS CONCERNING OPERATOR LIPSCHITZ FUNCTIONS A. B. Aleksandrov∗
UDC 517.98
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. Bibliography: 8 titles.
0. Introduction Let Lip(F) denote the space of the Lipschitz functions on a closed subset F of the complex plane C, i.e., Lip(F) is the set of the functions f defined on F such that |f (z2 ) − f (z1 )| ≤ C|z2 − z1 |
(0.1)
for all z1 , z2 ∈ F. Let f Lip(F) denote the minimal constant C ≥ 0 satisfying (0.1). A continuous function f defined on F is said to be operator Lipschitz if there exists a constant C such that1 f (N2 ) − f (N1 ) ≤ CN2 − N1
(0.2)
for any normal operators N1 , N2 acting on a Hilbert space H with spectra in F. We denote by OL(F) the set of the operator Lipschitz functions on F. Let f OL(F) denote the minimal constant C ≥ 0 satisfying (0.2). If a function f is defined on a larger set than F, then for the sake of brevity we write f ∈ OL(F) and f OL(F) instead of f |F ∈ OL(F) and f |F OL(F) . We write f OL(F) = +∞ if f ∈ OL(F). We also use these agreements for other function spaces. Clearly, OL(F) ⊂ Lip(F) and · Lip(F) ≤ · OL(F) . It is known that the equality OL(F) = Lip(F) is fulfilled for finite sets F only. We are interested in functions f ∈ OL(F) satisfying the equality f Lip(F) = f OL(F) . We consider mainly the case where F = R (in this case the normal operators N1 and N2 in (0.2) are self-adjoint). We consider also the case where F = T (then the corresponding normal operators are unitary). A continuous function on a closed set F is said to be commutator Lipschitz if there exists a constant C such that f (N )R − Rf (N ) ≤ CN R − RN
(0.3)
for any normal operator N on a Hilbert space H with spectrum in F and any bounded operator R on the same Hilbert space H. We denote by CL(F) the set of commutator Lipschitz functions on F. The smallest constant C ≥ 0 satisfying (0.3) is denoted by f CL(F) . ∗ St.Petersburg Department of Steklov Mathematical Institute, St.Petersburg, Russia, e-mail: [email protected]. 1In this paper we consider the operator norms only. Thus, if T is an operator, then T denotes the operator norm of T .
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 480, 2019, pp. 26–47. Original article submitted August 26, 2019. 176 1072-3374/20/2512-0176 ©2020 Springer Science+Business Media, LLC
Below we present several known properties of the operator Lipschitz and commutator Lipschitz functions. We refer t
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