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Invariant subspaces in the polydisk. (English) Zbl 0609.47012

From the introduction: ”(The) explicit characterization [due to A. Beurling, Acta Math. 81, 225-238 (1949; Zbl 0034.213)] of the invariant subspaces of the unilateral shift, in terms of the inner-outer factorization has had a major impact.” ”(But) most results (on the corresponding problem for \(H^ 2\) of the polydisk) have gone unpublished, and many results are counterexamples.” ”In this note, we eschew the goal and instead investigate equivalence of invariant subspaces, under joint unitary equivalence. Put another way, we shall consider the problem of characterizing the submodules of the Hardy space over the polydisk algebra.” ”We believe that this may be a worthwhile approach to this problem.”
Reviewer: Y.Kato

MSC:

47A15 Invariant subspaces of linear operators
46E20 Hilbert spaces of continuous, differentiable or analytic functions
42B30 \(H^p\)-spaces
46J15 Banach algebras of differentiable or analytic functions, \(H^p\)-spaces

Citations:

Zbl 0034.213
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