On the index of invariant subspaces in Hilbert spaces of vector-valued analytic functions

Let H be a Hilbert space of analytic functions on the unit disc D with ‖ M z ‖ ⩽ 1 , where M z 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 lim k → ∞ ‖ M z k f ‖ ≠ 0 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 C n -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)].