Universal Toeplitz operators on the Hardy space over the polydisk

2020/09/14 by Marcos S. Ferreira, Ferreira, Marcos, S. Waleed Noor +1
Mathematics · #Advanced Harmonic Analysis Research #Algebraic and Geometric Analysis #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory

paper · pdf · doi:10.48550/arxiv.2009.06751

openalex publication_date 2020/09/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The Invariant Subspace Problem (ISP) for Hilbert spaces asks if every bounded linear operator has a non-trivial closed invariant subspace. Due to the existence of universal operators (in the sense of Rota), the ISP may be solved by describing the invariant subspaces of these operators alone. We characterize all anaytic Toeplitz operators T_ϕ on the Hardy space H²(\mathbbDⁿ) over the polydisk \mathbbDⁿ for n>1 whose adjoints satisfy the Caradus criterion for universality, that is, when T_ϕ^* is surjective and has infinite dimensional kernel. In particular if ϕ in a non-constant inner function on \mathbbDⁿ, or a polynomial in the ring ℂ[z₁,…,z_n] that has zeros in \mathbbDⁿ but is zero-free on \mathbbTⁿ, then T_ϕ^* is universal for H²(\mathbbDⁿ). The analogs of these results for n=1 are not true.

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