Operator Lipschitz functions and model spaces

A. B. Aleksandrov

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 41, Tome 416 (2013), pp. 5-58

Résumé

Let $H^\infty$ denote the space of bounded analytic functions on the upper half plane $\mathbb C_+$. We prove that each function in the model space $H^\infty\cap\Theta\overline{H^\infty}$ is an operator Lipschitz function on $\mathbb R$ if and only if the inner function $\Theta$ is a usual Lipschitz function, i.e., $\Theta'\in H^\infty$. Let $(\mathrm{OL})'(\mathbb R)$ denote the set of all functions $f\in L^\infty$ whose antiderivative is operator Lipschitz on the real line $\mathbb R$. We prove that $H^\infty\cap\Theta\overline{H^\infty}\subset(\mathrm{OL})'(\mathbb R)$ if $\Theta$ is a Blaschke product with the zeros satisfying the uniform Frostman condition. We deal also with the following questions. When does an inner function $\Theta$ belong to $(\mathrm{OL})'(\mathbb R)$? When does each divisor of an inner function $\Theta$ belong to $(\mathrm{OL})'(\mathbb R)$? As an application, we deduce that $(\mathrm{OL})'(\mathbb R)$ is not a subalgebra of $L^\infty(\mathbb R)$. Another application is related to a description of the sets of discontinuity points for the derivatives of the operator Lipschitz functions. We prove that a set $\mathcal E$, $\mathcal E\subset\mathbb R$, is a set of discontinuity points for the derivative of an operator Lipschitz function if and only if $\mathcal E$ is an $F_\sigma$ set of first category. A considerable proportion of the results of the paper are based on a sufficient condition for operator Lipschitzness which was obtained by Arazy, Barton and Friedman. We give also a sufficient condition for operator Lipschitzness which is sharper than the Arazy–Barton–Friedman condition.