Ordered analytic Hilbert spaces over the unit disk

Shengzhao Hou ¹   ; Shuyun Wei ¹

¹ Department of Mathematics Suzhou University Suzhou, Jiangsu, 215006, P.R. China

Studia Mathematica, Tome 185 (2008) no. 2, pp. 127-142

Résumé

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 $X$.

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DOI : 10.4064/sm185-2-2

Keywords: analytic function ring hol mathbb unit disk mathbb say preceq there exist leq whenever hilbert space contained hol mathbb called ordered hilbert space preceq imply note mainly study connection between ordered analytic hilbert space its reproducing kernel consider invariant subspace whole space nbsp similar nbsp