Two theorems on star-invariant subspaces of BMOA

Abstract

For an inner function 0, let K 2 θ := H 2 e θH 2 and K *θ := K 2 θ n BMO. Two theorems are proved. The first of these provides a criterion for a coanalytic Toeplitz operator to map K *θ into a given space X, under certain assumptions on the latter. In particular, many natural smoothness spaces are eligible as X. As a consequence, for such spaces one has K *θ ⊂ X whenever θ ∈ X. The second theorem concerns the relationship between K *θ and its counterpart K *θ , defined as the image of K *θ under the natural involution f → zfθ. Specifically, it is proved that the inner factors associated with the two classes are the same if and only if θ is a Blaschke product whose zeros satisfy the so-called uniform Frostman condition.