With φ an inner function and $M_{φ}$ 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.
Department of Mathematics, Computer Science and Statistics, SUNY at Oneonta, Oneonta, NY 13820, U.S.A.