On imbedding theorems for coinvariant subspaces of the shift operator.~II

A. B. Aleksandrov

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 27, Tome 262 (1999), pp. 5-48

Résumé

For every inner function $\Theta$, we put $\Theta^*(H^2)\overset{\text{def}}=H^2\ominus\Theta H^2$, and $\Theta^*(H^p)\overset{\text{def}}=\operatorname{clos}_{H^p}(H^p\cap\Theta^*(H^2))$ for $p\ne 2$. Denote $\mathscr C_p(\Theta)=\{\mu\in C(\overline{\mathbb D}):\Theta^*(H^p)\subset L^p(|\mu|)\}$. An inner function $\Theta$ is said to be one-component if the set $\{z\in\mathbb D:|\Theta(z)|\varepsilon\}$ is connected for some $\varepsilon\in(0,1)$. A series of criteria for that are obtained. For example, $\Theta$ is one-component if and only if $\mathscr C_p(\Theta)$ does not depend on $p\in(0,+\infty)$. Moreover, there is a criterion in terms of the reproducing kernel of $\Theta^*(H^2)$. The set $\mathscr C_p(\Theta)$ is described in the case where $\Theta$ is a Blaschke product of special form. This description implies that the set of all $p$ such that a given measure $\mu$ belongs to $\mathscr C_p(\Theta)$ may have any finite or infinite number of connected component. The following examples of interpolating Blaschke products $\Theta$ and positive measures $\mu$ are constructed: (1) $\Theta^*(H^1)\subset L^1(\mu)$ and $\Theta^*(H^2)\subset L^2(\mu)$ but $\Theta^*(H^p)\not\subset L^p(\mu)$ for any $p\in(1,2)$; (2) $\Theta^*(H^p)\subset L^p(\mu)$ if and only if $p=\frac1n$, where $n$ is a positive integer; (3) $\Theta^*(H^p)\subset L^p(\mu)$ if and only if $p\ne\frac1n$, where $n$ is a positive integer.