2025/10/22 by David Békollè, Békollè, David, Hugues Olivier Défo +3
Mathematics · #46B50 #46E40 #46E50 #47B35 #47C15 #Algebraic and Geometric Analysis #Classical Analysis and ODEs (math.CA) #Complex Variables (math.CV) #FOS: Mathematics #Functional Analysis (math.FA) #Geometry and complex manifolds #Holomorphic and Operator Theory #Operator Algebras (math.OA) #Primary: 32A36 #Secondary: 32A25
paper · pdf · doi:10.48550/arxiv.2510.19589
openalex publication_date 2025/10/22 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28
As a class of compact operators on the ℓ2-valued Bergman space A2α(\mathbb Bn, ℓ2) on the unit ball \mathbb Bn, we study Toeplitz operators with BMO1α(\mathbb Bn, \mathcal L(ℓ2)) operator-valued symbols. First, we describe a method of restriction to a finite dimension which allows us to apply earlier results of Rahm and Wick; then we exhibit an explicit example of a compact Toeplitz operator on A2α(\mathbb Bn, ℓ2). Secondly, we apply two sufficient conditions for compactness established by Rahm in infinite dimension. The first condition is in terms of the Toeplitz algebra \mathcal T_L^∞fin, the second one is in terms of sufficiently localized operators and is implied by the first condition. To get the second condition, we additionally assume that the symbol and its adjoint belong to BMO1α(\mathbb Bn, \mathcal L(ℓ2, ℓ1)). Finally, inspired by Xia and Sadeghi-Zorboska, we ask the question of the validity of the reverse implication.