Commuting Toeplitz operators on Cartan domains of type IV and moment maps

2022/05/13 by Raúl Quiroga-Barranco, Quiroga-Barranco, Raul, Monyrattanak Seng +1
Mathematics · #53D20 #Algebraic and Geometric Analysis #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Primary 47B35 #Secondary 22D10 #Symplectic Geometry (math.SG)

paper · pdf · doi:10.48550/arxiv.2205.06786

openalex publication_date 2022/05/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let us consider, for n ≥ 3, the Cartan domain D_n^IV of type IV. On the weighted Bergman spaces A²_λ(D_n^IV) we study the problem of the existence of commutative C^*-algebras generated by Toeplitz operators with special symbols. We focus on the subgroup SO(n) × SO(2) of biholomorphisms of D_n^IV that fix the origin. The SO(n) × SO(2)-invariant symbols yield Toeplitz operators that generate commutative C^*-algebras, but commutativity is lost when we consider symbols invariant under a maximal torus or under SO(2). We compute the moment map μ^SO(2) for the SO(2)-action on D_n^IV considered as a symplectic manifold for the Bergman metric. We prove that the space of symbols of the form a = f ∘ μ^SO(2), denoted by L^∞(D_n^IV)^μ^SO(2), yield Toeplitz operators that generate commutative C^*-algebras. Spectral integral formulas for these Toeplitz operators are also obtained.

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