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Zapiski Nauchnykh Seminarov LOMI, 1986, Volume 149, Pages 38–51
(Mi znsl4925)
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This article is cited in 9 scientific papers (total in 9 papers)
Imbedding theorems for invariant subspaces of backward shift operator.
A. L. Vol'berg, S. R. Treil'
Abstract:
For subspaces $K_\theta^p=H^p\cap\theta\bar H^p_0$, $\theta$ being an inner function in the unit disc $\mathbb D$, we find conditions on a measure in $\operatorname{clos}\mathbb D$ ensuring the imbedding $K_\theta^p\subset L^p(\mu)$, $0<p<+\infty$. The main result claims that $K_\theta^p\subset L^p(\mu)$ if there are positive constants $\varepsilon$ and $c$ such that $\mu(\Delta)\leqslant c\cdot r_\Delta$ for every
disc $\Delta$ of radius $r_\Delta$ centered on $\mathbb T$ and such that $|\theta(z)|<\varepsilon$ for some $z\in\Delta$. Cohn's criterion for the imbedding $K_\theta^2\subset L^2(\mu)$ is obtained as a corollary. It is also shown that a necessary and sufficient condition for $K_\theta^p\subset L^p(\mu)$ must depend on $p$.
Citation:
A. L. Vol'berg, S. R. Treil', “Imbedding theorems for invariant subspaces of backward shift operator.”, Investigations on linear operators and function theory. Part XV, Zap. Nauchn. Sem. LOMI, 149, "Nauka", Leningrad. Otdel., Leningrad, 1986, 38–51
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https://www.mathnet.ru/eng/znsl4925 https://www.mathnet.ru/eng/znsl/v149/p38
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