# Journal of Operator Theory

Volume 50, Issue 2, Fall 2003 pp. 411-422.

Spectral density for multiplication operators with applications to factorizations of $L^1$ functions**Authors**: Isabelle Chalendar (1) and Jonathan R. Partington (2)

**Author institution:**(1) Institut Girard Desargues, UFR de Mathematiques, Universite Claude Bernard Lyon 1, 69622 Villeurbanne Cedex, France

(2) School of Mathematics, University of Leeds, Leeds LS2 9JT, UK

**Summary:**We give explicit formulae for the spectral density corresponding to $b(T)$ in terms of that associated with $T$, when $b$ is a finite Blaschke product and $T$ is an absolutely continuous contraction. As an application we obtain a decomposition of $L^1$ functions in terms of Hardy class functions. We use the decomposition of the spectral density corresponding to a particular multiplication operator in order to give a constructive proof of the fact that the classes $\AA_{m,n}$ (occurring in dual algebra theory) are all distinct.

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