# PREPRINT 23/2004

Vladimir Kapustin and Alexei Poltoratski

## BOUNDARY BEHAVIOR IN VECTOR-VALUED STAR-INVARIANT SUBSPACES

This preprint was accepted December 26, 2004

ABSTRACT:
In [1], D.~Clark considered certain families of positive singular
measures on the unit circle associated to inner functions $\theta$
in the unit disk. These measures were shown to be the spectral measures
of unitary rank-one perturbations of the model operator acting on the
(scalar) model space $K_\theta$, a subspace of the  Hardy space $H^2$.
In particular, it was shown that  $K_\theta$ can be mapped unitarily
onto $L^2(\sigma)$ for any of such measures $\sigma$. Later  it was
proved [2] that this mapping, which could be viewed as a generalization
of the classical Fourier transform, actually takes functions from
$K_\theta$ to their angular boundary values, which exist $\sigma$@-almost
everywhere for any such $\sigma$. In this paper we present an analog of
the above-mentioned results for vector-valued model spaces.
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