Ordered analytic Hilbert spaces over the unit disk
Abstract. Let f , g be in the analytic function ring Hol(D) over the unit disk D. We say that f g if there exist M > 0 and 0 < r < 1 such that |f(z)| ≤ M |g(z)| whenever r < |z| < 1. Let X be a Hilbert space contained in Hol(D). Then X is called an ordered Hilbert space if f g and g ∈ X imply f ∈ X. In this note, we mainly study the connection between an ordered analytic Hilbert space and its reproducing kernel. We also consider when an invariant subspace of the whole space X is similar to X.