Institution: | Department of Mathematics, Lund University, P.O. Box 118, S-221 00 Lund, Sweden ; Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996-1300 ; Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996-1300 |
Abstract: | Let be a Hilbert space of analytic functions on the open unit disc such that the operator of multiplication with the identity function defines a contraction operator. In terms of the reproducing kernel for we will characterize the largest set such that for each , the meromorphic function has nontangential limits a.e. on . We will see that the question of whether or not has linear Lebesgue measure 0 is related to questions concerning the invariant subspace structure of . We further associate with a second set , which is defined in terms of the norm on . For example, has the property that for all if and only if has linear Lebesgue measure 0. It turns out that a.e., by which we mean that has linear Lebesgue measure 0. We will study conditions that imply that a.e.. As one corollary to our results we will show that if dim and if there is a such that for all and all we have , then a.e. and the following four conditions are equivalent: (1) for some , (2) for all , , (3) has nonzero Lebesgue measure, (4) every nonzero invariant subspace of has index 1, i.e., satisfies dim . |