2021/03/31 by Amit Maji, Maji, Amit, Sankar T R +1
Mathematics · #32A10 #32A38 #32A70 #46E15 #47A13 #47A48 #Complex Variables (math.CV) #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA) #math.CV #math.FA #math.OA #msc:32A10 #msc:32A38 #msc:32A70 #msc:46E15 #msc:47A13 #msc:47A48
paper · pdf · doi:10.48550/arxiv.2103.17102
19 pages, revised version
arxiv created 2021/08/17 · arxiv updated 2021/08/19
We obtain a complete characterization for doubly commuting mixed invariant subspaces of the Hardy space over the unit polydisc. We say a closed subspace Q of H2(\mathbbDn) is mixed invariant if M_zj(Q) ⊆ Q for 1 ≤ j ≤ k and M_zj^*(Q) ⊆ Q, k+1 ≤ j ≤ n for some integer k ∈ \1, 2, …, n-1 \. We prove that a mixed invariant subspace Q of H2(\mathbbDn) is doubly commuting if and only if Q = ΘH2(\mathbbDk) ⊗ Qθ1 ⊗ ⋯ ⊗ Q_θn-k, where Θ∈ H∞(\mathbbDk) is some inner function and Qθj is either a Jordan block H2(\mathbbD)\ominus θj H2(\mathbbD) for some inner function θj or the Hardy space H2(\mathbbD). Furthermore, an explicit representation for the commutant of an n-tuple of doubly commuting shifts as well as a representation for the commutant of a doubly commuting tuple of shifts and co-shifts are obtained. Finally, we discuss some concrete examples of mixed invariant subspaces.