# Journal of Operator Theory

Volume 62, Issue 1, Summer 2009 pp. 111-123.

Factorisation spatiale**Authors**: Gilles Cassier (1) Jean Esterle (2)

**Author institution:**(1) Universite de Lyon, Lyon, F-69003, France and Universite Lyon 1, Institut Camille Jordan, Villeurbanne cedex, F-69622, France, and CNRS, UMR5208

(2) Universite de Bordeaux, IMB, UMR 5251, 351 Cours de la Liberation, 33405 Talence Cedex, France

**Summary:**We are firstly interested in finding the best possible compressions for a polynomially bounded operator $T$ that belongs to the class $\mathbb{A}_{1,1}$ introduced by H.~Bercovici, C.~Foia\c s and C.~Pearcy in \cite{bfp1}. Then, we use these compressions in order to obtain spatial factorizations, with a single vector, for large classes of lower semicontinuous positive functions $f$, in the sense that there exists a vector $x$ in the Hilbert space $H$ such that $\widehat{f}(n)=\langle T^{-n}x| x\rangle $, for all negative integer numbers $n$.

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