Invariant subspaces for finite index shifts in Hardy spaces and the invariant subspace problem for finite defect operators

2024/11/04 by Bracci, Filippo, Gallardo-Gutiérrez, Eva A.
#Complex Variables (math.CV) #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.2411.01933

Abstract

Let \mathbb H be the finite direct sums of H²(\mathbb D). In this paper, we give a characterization of the closed subspaces of \mathbb H which are invariant under the shift, thus obtaining a concrete Beurling-type theorem for the finite index shift. This characterization presents any such a subspace as the finite intersection, up to an inner function, of pre-images of a closed shift-invariant subspace of H²(\mathbb D) under ``determinantal operators'' from \mathbb H to H²(\mathbb D), that is, continuous linear operators which intertwine the shifts and appear as determinants of matrices with entries given by bounded holomorphic functions. With simple algebraic manipulations we provide a direct proof that every invariant closed subspace of codimension at least two sits into a non-trivial closed invariant subspace. As a consequence every bounded linear operator with finite defect has a nontrivial closed invariant subspace.

Related

• A Beurling Theorem for almost-invariant subspaces of the shift operator
• Study of nearly invariant subspaces with finite defect in Hilbert spaces
• Invariant subspaces of perturbed backward shift
• Beurling and Model subspaces invariant under a universal operator
• Almost invariant subspaces of the shift operator on vector-valued Hardy\n spaces