[Paper Review] H. Bohr's theorem for bounded symmetric domains

Guy Roos|ArXiv.org|Dec 28, 2008

Holomorphic and Operator Theory14 references3 citations

TL;DR

This paper generalizes H. Bohr's classical theorem on holomorphic self-maps of the unit disc to bounded symmetric domains of arbitrary type, proving that the sum of absolute values of Taylor coefficients converges to less than 1 within a ball of radius 1/3 in the spectral norm. The proof uses invariant theory and Jordan triple systems, avoiding case-by-case classification, and establishes the sharpness of the 1/3 bound for all bounded circled symmetric domains.

ABSTRACT

A theorem of Harald Bohr (1914) states that if f is a holomorphic map from the unit disc into itself, then the sum of absolute values of its Taylor expansion is less than 1 for |z|<1/3. The bound 1/3 is optimal. This result has been extended in a suitable sense by Liu Taishun and Wang Jianfei (2007) to the bounded complex symmetric domains of the four classical series, and to polydiscs. The result of Liu and Wang may be generalized to all bounded symmetric domains, with a proof which does not depend on classification.

Motivation & Objective

• To extend H. Bohr's 1914 theorem—originally for holomorphic maps from the unit disc to itself—beyond the complex unit disc to all bounded symmetric domains.
• To establish a uniform bound of 1/3 in the spectral norm for the convergence of the sum of absolute values of Taylor coefficients of holomorphic self-maps.
• To provide a classification-independent proof that avoids case-by-case analysis over the four classical series of bounded symmetric domains.
• To generalize the operator norm identity for automorphisms of bounded symmetric domains, showing ||dφ(u)||_Ω = 1/(1 - ||u||_Ω²) for any u in the domain.
• To demonstrate that the constant 1/3 is optimal, meaning the inequality fails for any larger radius.

Proposed method

• Uses the spectral norm ||·||_Ω associated with the Jordan triple system structure of bounded symmetric domains.
• Applies the Minkowski norm and Bergman operator theory to define the spectral norm as ||x||_Ω = λ₁, the largest eigenvalue in the spectral decomposition of x.
• Employs the simultaneous Peirce decomposition with respect to a frame (c₁,…,cᵣ) to analyze the structure of the Jordan triple system.
• Derives the key identity ||dφ(u)||_Ω = 1/(1 - ||u||_Ω²) for automorphisms φ fixing f(0), using the Bergman operator B(x,x) and its spectral properties.
• Applies the Taylor expansion f(z) = ∑fₖ(z) in homogeneous polynomials and bounds the sum ∑||Dφ(f(0))·fₖ(Z)||_Ω / ||Dφ(f(0))||_Ω.
• Uses the invariance of the spectral norm under automorphisms and the positivity of the Hermitian Jordan triple system to prove convergence within ||Z||_Ω < 1/3.

Experimental results

Research questions

• RQ1 Can Bohr’s theorem on the unit disc be generalized to all bounded symmetric domains without relying on classification?
• RQ2 What is the optimal radius r such that ∑|aₖzᵏ| ≤ 1 for all holomorphic self-maps f: Ω → Ω, where Ω is a bounded symmetric domain?
• RQ3 Does the operator norm of the derivative of an automorphism φ at a point u in Ω satisfy ||dφ(u)||_Ω = 1/(1 - ||u||_Ω²) for all bounded circled symmetric domains?
• RQ4 Is the constant 1/3 in Bohr’s theorem sharp for all bounded symmetric domains, including those not of classical type?
• RQ5 Can the spectral norm on the tangent space be used to define a universal convergence radius for Taylor series of holomorphic self-maps?

Key findings

• The paper proves that for any bounded symmetric domain Ω, if f: Ω → Ω is holomorphic and f(z) = ∑fₖ(z) is its Taylor expansion, then ∑||Dφ(f(0))·fₖ(Z)||_Ω / ||Dφ(f(0))||_Ω < 1 for all Z with ||Z||_Ω < 1/3.
• The bound 1/3 is optimal: for any r > 1/3, there exists a holomorphic self-map f: Ω → Ω such that the sum of absolute values exceeds 1 at some point with ||Z||_Ω = r.
• The operator norm identity ||dφ(u)||_Ω = 1/(1 - ||u||_Ω²) holds for all bounded circled symmetric domains, not just classical domains, via a classification-free proof.
• The spectral norm ||x||_Ω is defined via the spectral decomposition of x ∈ V, where x = ∑λᵢcᵢ with λ₁ > λ₂ > ... > 0, and ||x||_Ω = λ₁.
• The Bergman operator B(x,x) satisfies B(x,x) = ∑(1 - λᵢ²)(1 - λⱼ²)pᵢⱼ, and its spectral properties are used to derive the norm identity.
• The unit ball of the spectral norm ||·||_Ω coincides with the bounded symmetric domain Ω, and the Jordan triple system structure ensures the domain is bounded and circled.