Ordered analytic Hilbert spaces over the unit disk
Volume 185 / 2008
Studia Mathematica 185 (2008), 127-142
MSC: 46E22, 47B32.
DOI: 10.4064/sm185-2-2
Abstract
Let $f$, $g$ be in the analytic function ring ${\rm Hol}({\mathbb D})$ over the unit disk ${\mathbb D}$. We say that $f\preceq g$ if there exist $M>0$ and $0< r< 1$ such that $|f(z)|\leq M|g(z)|$ whenever $r< |z|< 1$. Let $X$ be a Hilbert space contained in ${\rm Hol}({\mathbb D})$. Then $X$ is called an ordered Hilbert space if $f\preceq g$ and $g\in X$ imply $f\in X$. 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 $X$.
