TY - JOUR

T1 - Operator Lipschitz Functions and Model Spaces

AU - Александров, Алексей Борисович

N1 - Publisher Copyright: © 2014, Springer Science+Business Media New York.

PY - 2014/10

Y1 - 2014/10

N2 - Let H∞ denote the space of bounded analytic functions on the upper half-plane ℂ+. We prove that each function in the model space H∞ ∩ Θ(Equation Presented) is an operator Lipschitz function on ℝ if and only if the inner function Θ is a usual Lipschitz function, i.e., Θ′ ∈ H∞. Let (OL)′(ℝ) denote the set of all functions f ∈ L∞ whose antiderivative is operator Lipschitz on the real line ℝ. We prove that H∞ ∩ Θ(Equation Presented) ⊂ (OL)′(ℝ) if Θ is a Blaschke product with zeros satisfying the uniform Frostman condition. We also deal with the following questions. When does an inner function Θ belong to (OL)′(ℝ)? When does each divisor of an inner function Θ belong to (OL)′(ℝ)? As an application, we deduce that (OL)′(ℝ) is not a subalgebra of L∞(ℝ). Another application is related to a description of the sets of discontinuity points for the derivatives of operator Lipschitz functions. We prove that a set ℰ,ℰ ⊂ ℝ, is a set of discontinuity points for the derivative of an operator Lipschitz function if and only if ℰ is an Fσ set of first category. A considerable proportion of the results of the paper is based on a sufficient condition for operator Lipschitzness which was obtained by Arazy, Barton, and Friedman. We also give a sufficient condition for operator Lipschitzness which is sharper than the Arazy–Barton–Friedman condition. Bibliography: 27 titles.

AB - Let H∞ denote the space of bounded analytic functions on the upper half-plane ℂ+. We prove that each function in the model space H∞ ∩ Θ(Equation Presented) is an operator Lipschitz function on ℝ if and only if the inner function Θ is a usual Lipschitz function, i.e., Θ′ ∈ H∞. Let (OL)′(ℝ) denote the set of all functions f ∈ L∞ whose antiderivative is operator Lipschitz on the real line ℝ. We prove that H∞ ∩ Θ(Equation Presented) ⊂ (OL)′(ℝ) if Θ is a Blaschke product with zeros satisfying the uniform Frostman condition. We also deal with the following questions. When does an inner function Θ belong to (OL)′(ℝ)? When does each divisor of an inner function Θ belong to (OL)′(ℝ)? As an application, we deduce that (OL)′(ℝ) is not a subalgebra of L∞(ℝ). Another application is related to a description of the sets of discontinuity points for the derivatives of operator Lipschitz functions. We prove that a set ℰ,ℰ ⊂ ℝ, is a set of discontinuity points for the derivative of an operator Lipschitz function if and only if ℰ is an Fσ set of first category. A considerable proportion of the results of the paper is based on a sufficient condition for operator Lipschitzness which was obtained by Arazy, Barton, and Friedman. We also give a sufficient condition for operator Lipschitzness which is sharper than the Arazy–Barton–Friedman condition. Bibliography: 27 titles.

UR - http://www.scopus.com/inward/record.url?scp=84922073790&partnerID=8YFLogxK

U2 - 10.1007/s10958-014-2057-y

DO - 10.1007/s10958-014-2057-y

M3 - Article

AN - SCOPUS:84922073790

VL - 202

SP - 485

EP - 518

JO - Journal of Mathematical Sciences

JF - Journal of Mathematical Sciences

SN - 1072-3374

IS - 4

ER -