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Invariant subspaces of $${\mathcal H}^ 2$$ of an annulus. (English) Zbl 0662.30035
A general study of shift-invariant subspaces of Hardy classes on multiply connected domains was first taken up by H. L. Royden long ago [Pac. J. Math. 134, 151-172 (1988; reviewed above)]. Following after Royden’s work, the author now presents in this interesting paper a complete classification of closed invariant subspaces of the Hardy class $$H^ 2(A)$$ on an annulus $$A=\{1<| z| <R\}$$- the simplest non-trivial multiply connected domain. The main result is the following theorem: Let M be a proper closed invariant subspace of $$H^ 2(A)$$. Suppose, for the sake of simplicity, that M has greatest common divisor 1. Then, there exist an integer m, an outer function h on $$\{| z| >1\}$$ and an inner function $$\psi$$ on the unit disk $$\Delta =\{| z| <1\}$$ such that $$M=z^ mhM_{\psi}$$, where $$M_{\psi}$$ consists of functions $$f\in H^ 2(A)$$ with the property: $$\psi (e^{i\theta})f(e^{i\theta})$$ are the boundary values of some element in $$H^ 2(\Delta)$$. The proof depends on two observations: first, the problem of determining invariant subspaces of $$H^ 2(A)$$ can be reduced to that of classifying closed subspaces F of $$H^ 2(\Delta)$$ which are weakly invariant with respect to backwards shift, meaning that $$g\in F$$ with $$g(0)=0$$ implies g(z)/z$$\in F$$, and, second, the latter problem can be solved. As for the latter problem he shows that each closed subspace F of $$H^ 2(\Delta)$$ which is weakly invariant with respect to backwards shift has the form $$\phi hN_{\psi}$$, $$\phi$$ and $$\psi$$ are inner functions on $$\Delta$$, h is an outer function on $$\Delta$$, and $$N_{\psi}$$ consists of functions $$f\in H^ 2(\Delta)$$ such that $$f(e^{i\theta})\overline{\psi (e^{i\theta})}$$ has an extension to an element of $$H^ 2(\{| z| >1\})$$. The proof needs some detailed analysis concerning the Szegö kernel functions. Having done this, he then notes that, for any closed invariant subspace M of $$H^ 2(A)$$, $$E=M\cap H^ 2(\{| z| >1\})$$ has the property: $$g\in E$$ and $$g(\infty)=0$$ imply zg(x)$$\in E$$. This is nothing but weakly invariance under backwards shift and the above result can be applied.
Reviewer: M.Hasumi

##### MSC:
 30D55 $$H^p$$-classes (MSC2000) 46J15 Banach algebras of differentiable or analytic functions, $$H^p$$-spaces
##### Keywords:
invariant subspace
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