Continuous and Compact Embeddings Between Star-invariant Subspaces

Abstract

Given an inner function θ on the upper half-plane ℂ₊, let $K_\theta ^p \mathop = \limits^{def} H^p \cap \theta \overline {H^p }$ be the corresponding star-invariant subspace of the Hardy space H _p = H _p (ℂ₊). It has been previously shown by the author that the (contin-uous) embedding relation K ^p _θ ⊂ K ^q _θ holds, for 1 < p < q < ∞, if and only if θ′ ∈ L ^∞(ℝ).In this paper, we first give an extended version of the above result and then establish a compactness criterion for the embedding operator involved. Namely, we prove that the inclusion map id : K ^p _θ ↪, K ^q _θ is compact if and only if θ′ ∈ C ₀(ℝ). Similar results are obtained on the embeddings K ^p _θ ⊂ C ₀(ℝ) and K ^p _θ ⊂ BMO.