$L^1$ factorizations, moment problems and invariant subspaces
Volume 167 / 2005
Studia Mathematica 167 (2005), 183-194
MSC: 47A15, 47A57, 30D55.
DOI: 10.4064/sm167-2-5
Abstract
For an absolutely continuous contraction $T$ on a Hilbert space ${{\mathcal H}}$, it is shown that the factorization of various classes of $L^1$ functions $f$ by vectors $x$ and $y$ in ${{\mathcal H}}$, in the sense that $\langle T^nx,y\rangle = \widehat f(-n)$ for $n \ge 0$, 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 $L^1$ factorizations and the moment sequences studied in the Atzmon–Godefroy method, from which further results on invariant subspaces are derived.
