An Invariant Subspace Theorem and Invariant Subspaces of Analytic Reproducing Kernel Hilbert Spaces - II

الأصل · EN

This paper is a follow-up contribution to our work [20] where we discussed some invariant subspace results for contractions on Hilbert spaces. Here we extend the results of [20] to the context of n-tuples of bounded linear operators on Hilbert spaces. Let T = (T₁,, Tₙ) be a pure commuting co-spherically contractive n-tuple of operators on a Hilbert space H and S be a non-trivial closed subspace of H. One of our main results states that: S is a joint T-invariant subspace if and only if there exists a partially isometric operator Π∈ B(H²ₙ(E), H) such that S = ΠH²ₙ(E), where H²ₙ is the Drury-Arveson space and E is a coefficient Hilbert space and Tᵢ Π= ΠMzᵢ, i = 1,, n. In particular, our work addresses the case of joint shift invariant subspaces of the Hardy space and the weighted Bergman spaces over the unit ball in Cⁿ.