[Paper Review] Hilbert-Schmidt Hankel operators with anti-holomorphic symbols on complete pseudoconvex Reinhardt domains

Mehmet Çeli̇k, Yunus E. Zeytuncu|arXiv (Cornell University)|Nov 16, 2015

Holomorphic and Operator Theory13 references3 citations

TL;DR

This paper establishes that on complete pseudoconvex Reinhardt domains in ℂ², there are no nonzero Hilbert-Schmidt Hankel operators with anti-holomorphic symbols, proving that such operators must vanish unless the symbol is constant. The result extends prior work on the unit ball and finite-type domains by leveraging pseudoconvexity and a key estimate from earlier work, while also constructing explicit examples of unbounded non-pseudoconvex domains that do admit nonzero Hilbert-Schmidt Hankel operators with anti-holomorphic symbols.

ABSTRACT

On complete pseudoconvex Reinhardt domains in $\mathbb{C}^2$, we show that there is no nonzero Hankel operator with an anti-holomorphic symbol that is Hilbert-Schmidt. We also present examples of unbounded non-pseudoconvex domains that admit nonzero Hilbert-Schmidt Hankel operators with anti-holomorphic symbols.

Motivation & Objective

• To characterize the symbols for which Hankel operators with anti-holomorphic symbols are Hilbert-Schmidt on complete pseudoconvex Reinhardt domains in ℂ².
• To extend previous results on the unit ball and finite-type domains by removing the finite type assumption.
• To demonstrate that pseudoconvexity is a key condition preventing nontrivial Hilbert-Schmidt Hankel operators.
• To construct explicit examples of unbounded non-pseudoconvex Reinhardt domains where such operators can be nonzero.
• To clarify the role of Bergman space structure and monomial basis orthogonality in determining Hilbert-Schmidt properties.

Proposed method

• The analysis uses the orthonormal basis of monomials {z^γ / c_γ} for A²(Ω) on complete Reinhardt domains, where c_γ² = ∫_Ω |z^γ|² dV.
• The Hankel operator H_̄f is defined as (I−P)(f̄g) on A²(Ω), with f ∈ A²(Ω), and its Hilbert-Schmidt norm is computed via the sum ∑_γ ||H_̄f(e_γ)||².
• The proof relies on a key estimate from prior work (equation (4) in [3]) and explicitly uses the pseudoconvexity of Ω to control growth of Bergman kernel coefficients.
• The authors analyze the sum S_α = ∑_k |c_{(k+α,k+α)}² / c_{(k,k)}² − c_{(k,k)}² / c_{(k−α,k−α)}²| for multi-indices α, showing divergence for nonzero α under pseudoconvexity.
• For counterexamples, the paper constructs unbounded non-pseudoconvex domains Ω₀ and Ω_k using regions X₁, X₂, X₃, and Bₘ, where Bergman spaces are spanned by (z₁z₂)^j.
• Explicit computation shows S_(1,1) ≈ ∑ 1/k² for large k, which converges, proving H_̄f is Hilbert-Schmidt for f(z) = z₁z₂ on Ω₀.

Experimental results

Research questions

• RQ1 On complete pseudoconvex Reinhardt domains in ℂ², when is a Hankel operator with anti-holomorphic symbol Hilbert-Schmidt?
• RQ2 Does the absence of nontrivial Hilbert-Schmidt Hankel operators on the unit ball extend to more general complete pseudoconvex Reinhardt domains without the finite type assumption?
• RQ3 Can unbounded non-pseudoconvex Reinhardt domains support nonzero Hilbert-Schmidt Hankel operators with anti-holomorphic symbols?
• RQ4 What structural properties of the Bergman space and monomial basis lead to convergence or divergence of the Hilbert-Schmidt norm of Hankel operators?
• RQ5 How does the canonical solution operator for the ∂̄-problem relate to the Hilbert-Schmidt property of Hankel operators on these domains?

Key findings

• On any complete pseudoconvex Reinhardt domain in ℂ², the only symbols f ∈ A²(Ω) for which H_̄f is Hilbert-Schmidt are constant functions.
• The proof relies on the divergence of the sum S_α for nonzero multi-indices α, which follows from the pseudoconvexity of Ω and the key estimate (4) from [3].
• For the unbounded non-pseudoconvex domain Ω₀ = X₁ ∪ X₂ ∪ X₃, the Bergman space A²(Ω₀) is infinite-dimensional and spanned by {(z₁z₂)^j}_{j=0}^∞.
• On Ω₀, the Hankel operator H_̄f with f(z) = z₁z₂ is Hilbert-Schmidt because S_(1,1) converges asymptotically like ∑ 1/k².
• On the finite-dimensional case Ω_k, the Hankel operator H_̄f is Hilbert-Schmidt for any f ∈ A²(Ω_k) due to the finite-dimensional domain of definition.
• The canonical solution operator for the ∂̄-problem on (0,1)-forms with holomorphic coefficients is not Hilbert-Schmidt on complete pseudoconvex Reinhardt domains, as it decomposes into a sum of non-Hilbert-Schmidt Hankel operators.