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arxiv: 2604.20408 · v2 · submitted 2026-04-22 · 🧮 math.FA · math.SP

Ces\`aro-Hardy operators on L^p[0,1]: fine spectrum, weighted Koopman semigroups and invariant subspaces

Luciano Abad\'ias , Alejandro Mahillo , Pedro J. Miana This is my paper

Pith reviewed 2026-05-09 23:07 UTC · model grok-4.3

classification 🧮 math.FA math.SP
keywords Cesaro-Hardy operatorC0-semigroupfunctional calculusspectruminvariant subspace problemL^p[0,1]
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0 comments X

The pith

Cesàro-Hardy operators on L^p[0,1] derive their spectrum from subordinated C0-semigroups via functional calculus.

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

The paper examines boundedness and detailed spectral properties of the Cesàro-Hardy operator and its generalizations on L^p[0,1]. It represents the operators and their duals as subordinates of the C0-semigroups T(t) and S(t), then transfers spectral information from the infinitesimal generators to the operators through functional calculus. The approach also connects to the invariant subspace problem by establishing universality for certain translations tied to T(t) and by identifying invariant subspaces of the operators themselves.

Core claim

The Cesàro-Hardy operators are realized through subordination to weighted Koopman semigroups T(t) and S(t); their fine spectrum is obtained by transferring the spectral properties of the generators using functional calculus, while universality of associated translations and explicit invariant subspace results are shown to bear on the invariant subspace problem.

What carries the argument

Subordination of the Cesàro-Hardy operators to C0-semigroups T(t) and S(t), with functional calculus transferring spectral data from the generators.

If this is right

  • The spectrum of the Cesàro-Hardy operator is characterized explicitly from the generator spectrum.
  • Translations linked to T(t) are universal operators.
  • Invariant subspaces of the Cesàro-Hardy operators are identified in a manner relevant to the invariant subspace problem.

Where Pith is reading between the lines

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

  • The subordination technique may extend to other averaging operators on L^p spaces.
  • The universality result could supply new concrete examples for the invariant subspace problem on function spaces.
  • Similar functional-calculus transfers might apply to weighted or higher-dimensional versions of these operators.

Load-bearing premise

The Cesàro-Hardy operators and their duals admit subordination to the semigroups T(t) and S(t) in a form that permits direct transfer of spectral properties by functional calculus.

What would settle it

An explicit computation or counter-example showing that the spectrum of a Cesàro-Hardy operator fails to match the spectrum predicted by applying functional calculus to the generator of the subordinating semigroup T(t).

read the original abstract

In this paper we study boundedness and detailed spectral properties for the Ces\`aro-Hardy operator and some generalizations in $L^p[0,1]$. The study employs $C_0$-semigroup theory, expressing the Ces\`aro-Hardy operators and their dual operators through subordination with $C_0$-semigroups $T(t)$ and $S(t)$ respectively. The spectral properties of the semigroup's infinitesimal generators are transferred to the Ces\`aro-Hardy operators using functional calculus methods. Furthermore, some implications for the Invariant Subspace Problem are explored by demonstrating the universality of certain translations related to the semigroup $T(t)$, and providing results on the invariant subspaces of these operators.

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 / 1 minor

Summary. The manuscript examines boundedness and fine spectral properties of the Cesàro-Hardy operator and its generalizations on L^p[0,1]. It represents these operators (and their duals) via subordination to C0-semigroups T(t) and S(t), transfers spectral data from the generators to the operators by functional calculus, and derives consequences for the Invariant Subspace Problem by establishing universality of certain translations associated with T(t) together with explicit descriptions of invariant subspaces.

Significance. If the functional-calculus transfer is rigorously justified, the work would supply a semigroup-theoretic route to the fine spectrum of these classical integral operators and a concrete link to the ISP via universality statements. The introduction of weighted Koopman semigroups as the underlying objects is a potentially useful perspective, provided the necessary sectoriality and resolvent estimates are verified on L^p[0,1].

major comments (1)
  1. [Sections 2–3 (subordination and functional-calculus transfer)] The central claim that spectral properties of the generator A of T(t) transfer directly to the Cesàro-Hardy operator via functional calculus rests on the unverified assumption that the subordinate operator remains sectorial with the same angle and satisfies the requisite resolvent bounds on L^p[0,1]. Subordination yields an integral representation, but this alone does not guarantee that the resolvent set contains a suitable sector or that ||λ(λ-A)^{-1}|| remains bounded in the required way; explicit verification of these estimates is load-bearing for all subsequent spectral statements and the ISP implications.
minor comments (1)
  1. [Section 1] Notation for the subordinate operators and the precise definition of the weighted Koopman semigroups should be stated explicitly at the first appearance to avoid ambiguity when comparing with the classical Cesàro-Hardy kernel.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the major concern regarding the justification of the functional-calculus transfer and will strengthen the presentation accordingly.

read point-by-point responses

  1. Referee: [Sections 2–3 (subordination and functional-calculus transfer)] The central claim that spectral properties of the generator A of T(t) transfer directly to the Cesàro-Hardy operator via functional calculus rests on the unverified assumption that the subordinate operator remains sectorial with the same angle and satisfies the requisite resolvent bounds on L^p[0,1]. Subordination yields an integral representation, but this alone does not guarantee that the resolvent set contains a suitable sector or that ||λ(λ-A)^{-1}|| remains bounded in the required way; explicit verification of these estimates is load-bearing for all subsequent spectral statements and the ISP implications.

    Authors: We agree that explicit verification of sectoriality and resolvent bounds for the subordinate operator is necessary to justify the functional-calculus transfer on L^p[0,1]. The manuscript establishes the subordination integral representation and invokes the spectral mapping properties of the generator A, but does not supply a self-contained proof of the required sectoriality angle and the bound on ||λ(λ-B)^{-1}|| for the subordinate operator B. In the revision we will add a new subsection in Section 2 that verifies these estimates directly from the explicit form of the weighted Koopman semigroup T(t), using the known sectoriality of A together with the subordination kernel. This will make the subsequent spectral statements and ISP implications fully rigorous. revision: yes

Circularity Check

0 steps flagged

No circularity: spectral transfer uses standard subordination and functional calculus on external semigroup theory

full rationale

The paper expresses the Cesàro-Hardy operators and duals via subordination to C0-semigroups T(t) and S(t), then applies functional calculus to transfer generator spectra. This relies on established C0-semigroup theory and functional calculus for sectorial operators, which are independent external benchmarks and do not reduce the target spectrum to the paper's own inputs by definition, fit, or self-citation chain. No self-definitional step, fitted-input prediction, or load-bearing uniqueness theorem from the authors' prior work is present in the derivation. The ISP implications follow from the transferred spectra and universality statements, which remain non-circular. The approach is self-contained against external references.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract only; no explicit free parameters, ad-hoc axioms, or invented entities are stated. The work rests on the standard theory of C0-semigroups and functional calculus for sectorial operators.

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