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arxiv: 2508.00666 · v2 · pith:MRTU4STAnew · submitted 2025-08-01 · 🧮 math.CV · math.FA

Eigenvalues for Infinitesimal Generators of Semigroups of Composition Operators

Maria Kourou , Eleftherios K. Theodosiadis , Konstantinos Zarvalis This is my paper

Pith reviewed 2026-05-19 01:33 UTC · model grok-4.3

classification 🧮 math.CV math.FA
keywords composition operatorsinfinitesimal generatorseigenvaluespoint spectrumKoenigs domainHardy spaceBergman spaceDirichlet space
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The pith

The point spectrum of the infinitesimal generator is characterized by containment relations and sufficient conditions depending on the holomorphic semigroup type and Koenigs domain geometry.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies eigenvalues of infinitesimal generators for semigroups of composition operators on Hardy, Bergman, and Dirichlet spaces, where the semigroups arise from families of holomorphic functions. It derives containment relations and sufficient conditions for the point spectrum that depend on the semigroup type, such as parabolic or other non-elliptic cases, and on the Euclidean geometry of the associated Koenigs domain. These results cover all non-elliptic semigroups for the Dirichlet space and parabolic semigroups for the Hardy and Bergman spaces, extending prior work on the hyperbolic case. A sympathetic reader would care because the findings connect the long-term dynamics of composition operators to concrete geometric features in the complex plane.

Core claim

Depending on the type of the holomorphic semigroup and the Euclidean geometry of its Koenigs domain, containment relations as well as sufficient conditions characterize the point spectrum of the induced infinitesimal generator on the Hardy spaces, Bergman spaces, and the Dirichlet space.

What carries the argument

The Koenigs domain of the holomorphic semigroup, whose Euclidean geometry together with the semigroup type supplies the containment relations and sufficient conditions for the point spectrum.

If this is right

  • For parabolic semigroups whose Koenigs domain meets certain geometric criteria, specific complex numbers lie in the point spectrum on Hardy and Bergman spaces.
  • The Dirichlet space admits the same geometry-based description of the point spectrum for every non-elliptic semigroup type.
  • The results supply explicit conditions that extend the known hyperbolic case to parabolic semigroups without requiring direct computation of the generator.
  • These relations link the planar shape of the Koenigs domain directly to the existence of eigenvalues for the induced operator.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same geometric criterion may apply to composition operators on additional spaces such as the Bloch space.
  • Numerical verification on explicit translation semigroups could test the sharpness of the geometry conditions.
  • The approach suggests that domain shape governs spectral features more than the particular choice of function space in these settings.
  • The characterizations could inform stability questions for dynamical systems generated by iterates of holomorphic maps.

Load-bearing premise

The spectral properties of the infinitesimal generator are determined by the semigroup type and the Euclidean geometry of its Koenigs domain in the non-elliptic and parabolic cases under consideration.

What would settle it

A concrete parabolic semigroup with a specified Koenigs domain geometry whose infinitesimal generator on the Hardy space has an eigenvalue outside the predicted containment would falsify the characterization.

Figures

Figures reproduced from arXiv: 2508.00666 by Eleftherios K. Theodosiadis, Konstantinos Zarvalis, Maria Kourou.

Figure 1
Figure 1. Figure 1: The inner/outer, lower and upper arguments, respectively. Theorem 1.4. Let (ϕt) be a parabolic semigroup of zero hyperbolic step in D which induces the semigroup of composition operators (Tt). Suppose that Ω is the Koenigs domain of (ϕt) and that σD is the point spectrum of the infinitesimal generator of (Tt). Then: (a) If θΩ = ΘΩ = 0, then σD = iS(0, π) ∪ {0}. (b) If θΩ = ΘΩ ∈ (0, π) (and hence θ − Ω = Θ−… view at source ↗
Figure 2
Figure 2. Figure 2: The domain Ω defined by ψΩ in Example 5.17, for x ∈ (0, 1/2] on the left and for x ∈ (1/2, 1) on the right. Example 5.17. Let x ∈ (0, 1) and consider the Koenigs domain Ω produced by the defining function ψΩ : (0, +∞) → [−∞, +∞) with ψΩ(y) = ( −∞, y ∈ (0, xπ] (y − xπ) cot(xπ) + log(y − xπ) − log sin(xπ), y ∈ (xπ, +∞). Clearly ψΩ is upper semi-continuous and Ω is well-defined as a Koenigs domain correspondi… view at source ↗
read the original abstract

We study the eigenvalues for infinitesimal generators of semigroups of composition operators acting on Hardy spaces, Bergman spaces, and the Dirichlet space. Such semigroups are induced by semigroups of holomorphic functions. Depending on the type of the holomorphic semigroup and the Euclidean geometry of its Koenigs domain, we find containment relations as well as sufficient conditions for the characterization of the point spectrum of the induced infinitesimal generator. For the Dirichlet space we study all types of non-elliptic semigroups whereas for the Hardy and Bergman spaces we work on parabolic semigroups extending the work of Betsakos in the hyperbolic case.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript studies the point spectrum of infinitesimal generators of semigroups of composition operators on Hardy, Bergman, and Dirichlet spaces. The central results consist of containment relations and sufficient conditions for eigenvalues, determined by the type of the underlying holomorphic semigroup (non-elliptic or parabolic) and the Euclidean geometry of its Koenigs domain. The Dirichlet-space analysis covers all non-elliptic cases, while the Hardy/Bergman analysis treats parabolic semigroups and extends earlier hyperbolic results.

Significance. If the derivations hold, the geometric criterion via Koenigs domains supplies a concrete mechanism for locating point spectra of these generators, extending the hyperbolic case treated by Betsakos to the parabolic setting and adding the Dirichlet space. This supplies falsifiable predictions about eigenvalue locations that depend only on domain geometry rather than on fitted parameters.

major comments (1)
  1. [§4.2, Theorem 4.3] §4.2, Theorem 4.3: the sufficient condition for 0 to lie in the point spectrum of the generator on the Bergman space is stated in terms of the boundary behavior of the Koenigs function; however, the proof sketch does not address whether the same geometric hypothesis is also necessary, which weakens the claimed characterization.
minor comments (2)
  1. [Theorem 3.1] The statement of the main containment relation in the parabolic Dirichlet-space case (Theorem 3.1) would benefit from an explicit comparison with the corresponding hyperbolic result of Betsakos to highlight the new geometric features.
  2. [§2 and §5] Notation for the Koenigs function and its derivative is introduced in §2 but used with slight variations in §5; a single consistent definition would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the positive overall assessment. We address the single major comment below and will incorporate clarifications in the revised version.

read point-by-point responses

  1. Referee: [§4.2, Theorem 4.3] §4.2, Theorem 4.3: the sufficient condition for 0 to lie in the point spectrum of the generator on the Bergman space is stated in terms of the boundary behavior of the Koenigs function; however, the proof sketch does not address whether the same geometric hypothesis is also necessary, which weakens the claimed characterization.

    Authors: We agree that Theorem 4.3 provides only a sufficient condition for 0 to belong to the point spectrum on the Bergman space, expressed via the boundary behavior of the Koenigs function, and that the proof establishes sufficiency but does not treat necessity. The manuscript statement and abstract refer to 'sufficient conditions for the characterization,' which may suggest a stronger result than intended. We will revise the theorem statement to make explicit that only sufficiency is claimed, add a remark noting that necessity is not addressed and may require separate arguments or additional hypotheses in the parabolic case, and adjust the abstract wording for precision. These changes preserve the main contribution while improving clarity. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper derives containment relations and sufficient conditions for the point spectrum of infinitesimal generators from the classification of holomorphic semigroups (non-elliptic, parabolic) and the Euclidean geometry of their Koenigs domains, acting on Hardy, Bergman, and Dirichlet spaces. These relations follow from standard operator-theoretic arguments on composition operators and semigroup theory, extending external prior results (e.g., Betsakos on hyperbolic cases) without reducing any central claim to a fitted parameter, self-definition, or self-citation chain. The setup treats domain geometry as an independent input that determines spectral properties, yielding a self-contained analysis against external benchmarks in complex analysis.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard domain assumptions in complex analysis regarding holomorphic semigroups and the geometry of Koenigs domains. No free parameters or invented entities are indicated.

axioms (1)
  • domain assumption Semigroups of holomorphic functions induce composition operator semigroups on Hardy, Bergman, and Dirichlet spaces whose spectral properties depend on semigroup type and Koenigs domain geometry.
    This is the foundational setup described in the abstract for deriving the containment relations and conditions.

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Reference graph

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