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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm185-2-2,
author = {Shengzhao Hou and Shuyun Wei},
title = {Ordered analytic Hilbert spaces over the unit disk},
journal = {Studia Mathematica},
volume = {187},
year = {2008},
pages = {127-142},
zbl = {1152.46016},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm185-2-2}
}
Shengzhao Hou; Shuyun Wei. Ordered analytic Hilbert spaces over the unit disk. Studia Mathematica, Tome 187 (2008) pp. 127-142. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm185-2-2/