Invariant Subspaces in Banach Spaces of Analytic Functions

Abstract

Abstract. We study the invariant subspace structure of the operator of multiplication by z, M., on a class of Banach spaces of analytic functions. For operators on Hilbert spaces our class coincides with the adjoints of the operators in the CowenDouglas class 3SX(Ü). We say that an invariant subspace Jt satisfies cod^"= 1 if zJt has codimension one in Jt'. We give various conditions on invariant subspaces which imply that cod Jt = 1. In particular, we give a necessary and sufficient condition on two invariant subspaces Jt, jV with cod Jt = cod jV= 1 so that their span again satisfies cod(^" v jV) = 1. This result will be used to show that any invariant subspace of the Bergman space L¡¡, p > 1, which is generated by functions in L1/, must satisfy cod Jf = I. For an invariant subspace Jt we then consider the operator S = M7 \JtL. Under some extra assumption on the domain of holomorphy we show that the spectrum of S coincides with the approximate point spectrum iff cod Jt = 1. Finally, in the last section we obtain a structure theorem for invariant subspaces with cod^= 1. This theorem applies to Dirichlet-type spaces.

Multiplication operators on the Bergman space via the Hardy space of the bidisk

“…Our main idea is to lift the Bergman shift up as the compression of a commuting pair of isometries on a nice subspace of the Hardy space of bidisk. This idea was used in studying the Hilbert modules by R. Douglas and V. Paulsen [12], operator theory in the Hardy space over the bidisk by R. Dougals and R. Yang [13], [37], [38] and [39]; the higher-order Hankel forms by S. Ferguson and R. Rochberg [10] and [11] and the lattice of the invariant subspaces of the Bergman shift by S. Richter [22].…”

Hilbert Spaces of Analytic Functions Between the Hardy and the Dirichlet Space

“…As it was pointed out in [3] this implies the fact that each nontrivial invariant subspace JÍ of Mz has the codimension one property that is, (z -A)^# is a closed subspace of JÍ having codimension 1 in Jf, for every X e U. This follows from results obtained by S. Richter in [2] and Theorem 3. …”

Beurling type theorem on the Bergman space via the Hardy space of the bidisk

“…Let M be an invariant subspace of B and M denote the lifting of M. It was shown in [14] that M is an invariant subspace of T z . For completeness we include a proof.…”

“…operator theory in the Hardy space over the bidisk by R. Douglas and R. Yang [5], [19], [20] and [21]; the higher-order Hankel forms by S. Ferguson and R. Rochberg [7] and [8] and the lattice of the invariant subspaces of the Bergman shift by S. Richter [14].…”

“…This paper is organized as follows. In Section 2, as in [14] we lift each invariant subspace of B as an invariant subspace of the isometry T z , do the Wold decomposition and identify the the wandering subspace. In Section 3, using the structure of the wandering space and establishing an identity (in Step 8 in the proof of Theorem 3.1) we give a proof of Aleman, Richter and Sundberg theorem [1].…”