zbMATH — the first resource for mathematics

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

30D55 \(H^p\)-classes (MSC2000)
46J15 Banach algebras of differentiable or analytic functions, \(H^p\)-spaces
Full Text: DOI