Invariant subspaces for Toeplitz operators

V. V. Peller

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part XII, Tome 126 (1983), pp. 170-179

Résumé

The article is devoted to the invariant subspace problem for Toeplitz operators. Let $\Gamma$ be a Lipschitz arc on the complex plane, $f$ be a non-constant continuous function on the unit circle. If there exists an open disc $D$ such that $f(\mathbb T)\cap\Gamma\cap D\ne\varnothing$, $f(\mathbb T)\cap(\bar D\setminus\Gamma)\ne\varnothing$ and if the modulus of continuity $\omega_f$ of $f$ satisfies the inequality $$ \int\limits_\bigcirc\frac{\omega_f(t)}{t\log\frac1t}\,dt<+\infty, $$ then non-trivial hyperinvariant subspaces for the Toeplitz operator $T_f$ on the Hardy class $H^2$ are proved to exist. For the proof of this result the Lubich–Matsaev theorem is used.