Invariant subspaces in VMOA and BMOA
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\}.
spaces of functions of bounded mean oscillation
space of functions of vanishing mean oscillation
scientific article; zbMATH DE number 682536