For an absolutely continuous contraction T on a Hilbert space 𝓗, it is shown that the factorization of various classes of L¹ functions f by vectors x and y in 𝓗, in the sense that ⟨Tⁿx,y⟩ = f̂(-n) for n ≥ 0, implies the existence of invariant subspaces for T, or in some cases for rational functions of T. One of the main tools employed is the operator-valued Poisson kernel. Finally, a link is established between L¹ factorizations and the moment sequences studied in the Atzmon-Godefroy method, from which further results on invariant subspaces are derived.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm167-2-5,
author = {Isabelle Chalendar and Jonathan R. Partington and Rachael C. Smith},
title = {L$^1$ factorizations, moment problems and invariant subspaces},
journal = {Studia Mathematica},
volume = {166},
year = {2005},
pages = {183-194},
zbl = {1081.47005},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm167-2-5}
}
Isabelle Chalendar; Jonathan R. Partington; Rachael C. Smith. L¹ factorizations, moment problems and invariant subspaces. Studia Mathematica, Tome 166 (2005) pp. 183-194. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm167-2-5/