Invariant subspaces in VMOA and BMOA
type: Item
instance of: scholarly article
title: Invariant subspaces in VMOA and BMOA
zbMATH Open document ID: 0854.46024
author: D. C. Singh
author: U. N. Singh
published in: Michigan Mathematical Journal
publication date: 1994-11-06T00:00:00Z
review text: Classical invariant subspace theorems for Hardy spaces are extended in this paper to spaces of functions of bounded mean oscillation. BMO is the Banach space of (Lebesgue) integrable functions of bounded mean oscillation on the circle \(T\). The closure in BMO of the continuous function on \(T\) is VMO, the space of functions of vanishing mean oscillation. BMOA and VMOA denote their intersections with the Hardy space \(H^1\). Both BMOA and VMOA are invariant under the operator \(S\) of multiplication by the coordinate function \(z\). The principal results in the paper, Theorems A and C, describe the closed (in norm for VMOA, in the weak-star topology for BMOA) subspaces of VMOA and BMOA that are invariant under \(S\); here the weak-star topology is that which BMOA enjoys as the dual space of \(H^1\). Theorem A asserts that the nontrivial proper closed \(S\)-invariant subspaces of VMOA are precisely the spaces of the form \(M=IN\) where \(I\) is a unique (up to multiplication by a scalar of modulus 1) inner function and \(N\) is a unique subspace of VMOA; \(N\) is norm dense in VMOA, and \(N=\text{VMOA}\) if and only if \(I\) is a finite Blaschke product. It is observed that every inner function \(I\) arises in this way with exactly one \(N\). Theorem C and the corresponding observation are the same as Theorem A and its observation, but with BMOA in place of VMOA, and the weak-star topology in place of the norm topology. The proofs use ``classical'' results as well as results of \textit{J. M. Anderson} [Comment. Math. Helv. 54, 309-317 (1979; Zbl 0411.30020)] and \textit{D. Stegenga} [Am. J. Math. 98, 573-589 (1976; Zbl 0335.47018)], and pass through \(S^*\), the adjoint of \(S\) on \(H^1\) (which is the dual space of VMOA). \{Reviewer's remark: Equation (1.1) needs an obvious modification\}.
Mathematics Subject Classification ID: 46E15
Mathematics Subject Classification ID: 46J15
Mathematics Subject Classification ID: 47A15
Mathematics Subject Classification ID: 32A37
Mathematics Subject Classification ID: 30H05
Mathematics Subject Classification ID: 32A35
Mathematics Subject Classification ID: 42B30
zbMATH DE Number: 682536
zbMATH Keywords: invariant subspace theorems
zbMATH Keywords: Hardy spaces
zbMATH Keywords: spaces of functions of bounded mean oscillation
zbMATH Keywords: BMO
zbMATH Keywords: space of functions of vanishing mean oscillation
zbMATH Keywords: Blaschke product
reviewed by: Stuart J. Sidney
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1029004989
OpenAlex ID: W2086556454
DOI: 10.1307/MMJ/1029004989
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description: scientific article; zbMATH DE number 682536