$L^1$ factorizations, moment problems and invariant subspaces

Isabelle Chalendar ¹   ; Jonathan R. Partington ²   ; Rachael C. Smith ²

¹ Institut Girard Desargues, UFR de Mathématiques Université Claude Bernard Lyon 1 69622 Villeurbanne Cedex, France
² School of Mathematics University of Leeds Leeds LS2 9JT, U.K.

Studia Mathematica, Tome 167 (2005) no. 2, pp. 183-194

Résumé

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.

Détail

DOI : 10.4064/sm167-2-5

Keywords: absolutely continuous contraction hilbert space mathcal shown factorization various classes functions vectors mathcal sense langle rangle widehat n implies existence invariant subspaces cases rational functions main tools employed operator valued poisson kernel finally link established between factorizations moment sequences studied atzmon godefroy method which further results invariant subspaces derived