Growth estimates for generalized factors of $H^p$ spaces

Joseph A. Cima¹ ; Angeliki Kazas² ; Michael I. Stessin³

¹ Department of Mathematics University of North Carolina 346 Phillips Hall 3250 Chapel Hill, NC 27599-3250, U.S.A.
² Department of Mathematics, Computer Science, and Statistics SUNY at Oneonta Oneonta, NY 13820, U.S.A.
³ Department of Mathematics and Statistics SUNY at Albany Albany, NY 12222, U.S.A.

Studia Mathematica, Tome 158 (2003) no. 1, pp. 19-38

Résumé

With $\varphi $ an inner function and $M_{\varphi }$ the multiplication operator on a given Hardy space it is known that for any given function $f$ in the Hardy space we may use the Wold decomposition to obtain a factorization of the given $f$ (not the Riesz factorization). This new factorization has been shown to be useful in the study of commutants of Toeplitz operators. We study the smoothness of each factor of this factorization. We show in some cases that the factors lie in the same Hardy space (or smoothness class) as the given function $f$. We also construct an example to show that there are bounded, holomorphic functions which have factors that are not in a given Hardy $p$-space. Many of our results are produced by studying a natural class of positive measures associated to the given inner function.

Détail

DOI : 10.4064/sm158-1-3

Keywords: varphi inner function varphi multiplication operator given hardy space known given function hardy space may wold decomposition obtain factorization given riesz factorization factorization has shown useful study commutants toeplitz operators study smoothness each factor factorization cases factors lie hardy space smoothness class given function construct example there bounded holomorphic functions which have factors given hardy p space many results produced studying natural class positive measures associated given inner function