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

2013/10/03 by Jaydeb Sarkar, Sarkar, Jaydeb
Mathematics · #30H05 #46E22 #46M05 #46N99 #47A13 #47A15 #47A20 #47A45 #47B32 #47B38 #Algebraic and Geometric Analysis #Complex Variables (math.CV) #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Operator Algebras (math.OA) #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.1310.1014

openalex publication_date 2013/10/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This paper is a follow-up contribution to our work [20] where we discussed\nsome invariant subspace results for contractions on Hilbert spaces. Here we\nextend the results of [20] to the context of n-tuples of bounded linear\noperators on Hilbert spaces. Let T = (T₁, \…, T_n) be a pure commuting\nco-spherically contractive n-tuple of operators on a Hilbert space \H\nand \S be a non-trivial closed subspace of \H. One of our\nmain results states that: \S is a joint T-invariant subspace if and\nonly if there exists a partially isometric operator \Π \∈\n\B(H²_n(\E), \H) such that \S = \Π\nH²_n(\E), where H²_n is the Drury-Arveson space and \E is\na coefficient Hilbert space and T_i \Π = \Π M_zi, i = 1, \…, n. In\nparticular, our work addresses the case of joint shift invariant subspaces of\nthe Hardy space and the weighted Bergman spaces over the unit ball in\n\ℂⁿ.\n

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