Operator Lipschitz Functions and Model Spaces

Abstract

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 $$ \overline{H^{\infty }} $$ 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 $$ \overline{H^{\infty }} $$ ⊂ (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 F σ 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.