Linear graph transformations on spaces of analytic functions

Abstract

Let H be a Hilbert space of analytic functions with multiplier algebra M(H), and letM={(f,T1f,…,Tn−1f):f∈D} be an invariant graph subspace for M(H)(n). Here n≥2, D⊆H is a vector-subspace, Ti:D→H are linear transformations that commute with each multiplication operator Mφ∈M(H), and M is closed in H(n). In this paper we investigate the existence of non-trivial common invariant subspaces of operator algebras of the typeAM={A∈B(H):AD⊆D:ATif=TiAf∀f∈D}. In particular, for the Bergman space La2 we exhibit examples of invariant graph subspaces of fiber dimension 2 such that AM does not have any nontrivial invariant subspaces that are defined by linear relations of the graph transformations for M.