On the index of invariant subspaces in Hilbert spaces of vector-valued analytic functions |
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Authors: | Marcus Carlsson |
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Affiliation: | Department of Mathematics, Purdue University, 150 N. University Street, West Lafayette, IN 47907-2067, USA |
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Abstract: | Let H be a Hilbert space of analytic functions on the unit disc D with ‖Mz‖?1, where Mz denotes the operator of multiplication by the identity function on D. Under certain conditions on H it has been shown by Aleman, Richter and Sundberg that all invariant subspaces have index 1 if and only if for all f∈H, f?0 [A. Aleman, S. Richter, C. Sundberg, Analytic contractions and non-tangential limits, Trans. Amer. Math. Soc. 359 (7) (2007) 3369-3407]. We show that the natural counterpart to this statement in Hilbert spaces of Cn-valued analytic functions is false and prove a correct generalization of the theorem. In doing so we obtain new information on the boundary behavior of functions in such spaces, thereby improving the main result of [M. Carlsson, Boundary behavior in Hilbert spaces of vector-valued analytic functions, J. Funct. Anal. 247 (1) (2007)]. |
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Keywords: | Vector-valued analytic functions Nontangential limits Invariant subspaces Index |
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