Analytic Contractions, Nontangential Limits, and the Index of Invariant Subspaces

Let H be a Hilbert space of analytic functions on the open unit disc D such that the operator Mζ of multiplication with the identity function ζ defines a contraction operator. In terms of the reproducing kernel for H we will characterize the largest set ∆(H) ⊆ ∂D such that for each f, g ∈ H, g 6= 0 the meromorphic function f/g has nontangential limits a.e. on ∆(H). We will see that the question of whether or not ∆(H) has linear Lebesgue measure 0 is related to questions about the invariant subspace structure of Mζ . We further associate with H a second set Σ(H) ⊆ ∂D which is defined in terms of the norm on H. For example, Σ(H) has the property that ||ζnf || → 0 for all f ∈ H if and only if Σ(H) has linear Lebesgue measure 0. It turns out that ∆(H) ⊆ Σ(H) a.e., by which we mean that ∆(H) \ Σ(H) has linear Lebesgue measure 0. We will study conditions that imply that ∆(H) = Σ(H) a.e.. As one corollary to our results we will show that if dim H/ζH = 1 and if there is a c > 0 such that for all f ∈ H and all λ ∈ D we have || ζ−λ 1−λζ f || ≥ c||f ||, then ∆(H) = Σ(H) a.e. and the following four conditions are equivalent: (1) ||ζnf ||9 0 for some f ∈ H, (2) ||ζnf ||9 0 for all f ∈ H, f 6= 0, (3) ∆(H) has nonzero Lebesgue measure, (4) every nonzero invariant subspace M of Mζ has index 1, i.e. satisfies dim M/ζM = 1. 2000 Mathematics Subject Classification. Primary 47B32, 46E22; Secondary 30H05, 46E20.