[Paper Review] Multiplication operators on the Bergman space of bounded domains in C^d

Hansong Huang, Dechao Zheng|arXiv (Cornell University)|Nov 5, 2015

Holomorphic and Operator Theory14 references6 citations

TL;DR

This paper investigates multiplication operators on Bergman spaces over bounded domains in ℂ^d, establishing that the von Neumann algebra generated by such operators is finite-dimensional when the symbol is a holomorphic proper map with interior image. It generalizes single-variable results on commutants and reducing subspaces using admissible local inverses, analytic continuation, and L^2_a-removable sets, showing that the structure of the algebra depends on the geometry and function-theoretic properties of the symbol map in higher dimensions.

ABSTRACT

In this paper we study multiplication operators on Bergman spaces of high dimensional bounded domains and those von Neumann algebras induced by them via the geometry of domains and function theory of their symbols. In particular, using local inverses and $L^2_a$-removability, we show that for a holomorphic proper map $Φ=(ϕ_1, ϕ_2, \cdots , ϕ_d)$ on a bounded domain $Ω$ in $\mathbb{C}^{d}$, the dimension of the von Neumann algebra $\mathcal{V}^*(Φ,Ω) $ consisting of bounded operators on the Bergman space $L_a^2(Ω)$, which commute with both $ M_{ϕ_j}$ and its adjoint $M_{ϕ_j}^*$ for each $j$, equals the number of components of the complex manifold $\mathcal{S}_{Φ}= \{(z,w)\in Ω^2: Φ(z)=Φ(w),\, z ot\in Φ^{-1}(Φ(Z))\},$ where $Z$ is the zero variety of the Jacobian $JΦ$ of $ Φ.$ This extends the main result in \cite{DSZ} in high dimensional complex domains. Moreover we show that the von Neumann algebra $\mathcal{V}^*(Φ,Ω) $ may not be abelian in general although Douglas, Putinar and Wang \cite{DPW} showed that $\mathcal{V}^*(Φ,\mathbb{D})$ for the unit disk $\mathbb{D}$ is abelian.

Motivation & Objective

• To extend single-variable results on commutants and von Neumann algebras of multiplication operators to higher-dimensional bounded domains in ℂ^d.
• To investigate the structure of the von Neumann algebra 𝒱*(Φ,Ω) generated by multiplication operators and their adjoints on Bergman spaces.
• To determine conditions under which 𝒱*(Φ,Ω) is finite-dimensional or trivial, particularly in relation to the geometry of the domain and the analytic properties of the symbol map Φ.
• To explore the role of admissible local inverses, analytic continuation, and L^2_a-removable sets in characterizing the commutant and the von Neumann algebra.
• To demonstrate that the classical single-variable result (ϕ = ψ∘B implies {M_ϕ}′ = {M_B}′) fails in higher dimensions, providing a counterexample in ℂ^2.

Proposed method

• Define the von Neumann algebra 𝒱*(Φ,Ω) as the commutant of {M_ϕ_j,Ω, M_ϕ_j,Ω*} for a tuple Φ of holomorphic functions on a bounded domain Ω ⊂ ℂ^d.
• Introduce the concept of admissible local inverses of Φ, which are local holomorphic inverses whose analytic continuations yield bounded operators on L_a^2(Ω).
• Construct operators 𝒪_{[ρ]} on L_a^2(Ω) via summing h∘σ·Jσ over equivalence classes [ρ] of admissible local inverses, where h ∈ L_a^2(Ω).
• Use the Hartogs phenomenon and L^2_a-removable sets to analyze the domain of analytic continuation of local inverses and their boundedness.
• Apply the theory of holomorphic proper maps and their Jacobian determinants to relate the number of components of the Riemann surface of Φ to the dimension of 𝒱*(Φ,Ω).
• Construct explicit counterexamples (e.g., Φ(z₁,z₂) = (z₁²+z₂², z₁²z₂²) on ℂ^2 minus a small ball) to show that {M_Φ}′ ≠ {M_Ψ}′ even when Φ is a function of Ψ, violating the single-variable generalization.

Experimental results

Research questions

• RQ1 Under what geometric and analytic conditions on a bounded domain Ω ⊂ ℂ^d is the von Neumann algebra 𝒱*(Φ,Ω) finite-dimensional?
• RQ2 How do admissible local inverses and their analytic continuation relate to the structure of the commutant {M_Φ,Ω}′ and the von Neumann algebra 𝒱*(Φ,Ω)?
• RQ3 To what extent does the single-variable result {M_ϕ}′ = {M_B}′ for ϕ = ψ∘B generalize to higher dimensions?
• RQ4 Can a holomorphic proper map Φ on Ω be expressed as a function of another holomorphic proper map Ψ such that {M_Φ}′ = {M_Ψ}′ in higher dimensions?
• RQ5 What is the role of L^2_a-removable sets and the Hartogs phenomenon in determining the boundedness and commutativity of operators associated with local inverses?

Key findings

• The von Neumann algebra 𝒱*(Φ,Ω) is finite-dimensional if Ω satisfies properties (1) and (2) and Φ: Ω → ℂ^d is holomorphic on 𝒪(Ω) with image containing an interior point.
• The algebra 𝒱*(Φ,Ω) is generated by the operators 𝒪_{[ρ]} associated with admissible local inverses ρ of Φ.
• If the image of Φ has no interior point, then 𝒱*(Φ,Ω) may be infinite-dimensional, as shown in Example 4.5.
• In Example 7.4, removing a small closed ball from the unit ball or bidisk results in a domain Ω where 𝒱*(Φ,Ω) = ℂI, even though the commutant {M_Φ,Ω}′ contains 8 nontrivial operators.
• The classical single-variable result {M_ϕ}′ = {M_B}′ for ϕ = ψ∘B does not generalize to higher dimensions: Example 7.5 shows that even if Φ = h∘Ψ, it may not hold that {M_Φ}′ = {M_Ψ}′.
• The existence of a holomorphic proper map Ψ such that {M_Φ}′ = {M_Ψ}′ and Φ = h∘Ψ leads to a contradiction in Example 7.5, due to mismatched multiplicities of roots of Φ−Φ(λ).