arxiv: 2604.24296 · v1 · submitted 2026-04-27 · 🧮 math.FA

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H^infty--functional calculus for generators of semigroups that admit lower bounds

Benhard H. Haak, Peer Chr. Kunstmann

Pith reviewed 2026-05-07 17:42 UTC · model grok-4.3

classification 🧮 math.FA

keywords C0-semigroupsH infinity functional calculusUMD Banach spacesdilationtransferencelower boundssemigroup generators

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The pith

If one operator of a C0-semigroup on a UMD Banach space admits a lower bound, the negative generator has a bounded H infinity functional calculus.

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

The paper establishes that C0-semigroups on UMD Banach spaces satisfying a lower bound condition for at least one positive time operator possess generators with bounded H infinity functional calculi. The argument relies on dilating the given semigroup to a C0-group on an enlarged space via a recent construction, then transferring known calculus bounds for such groups back to the original generator using transference techniques available on UMD spaces. A reader would care because the result relaxes the usual sectoriality or analyticity hypotheses that dominate Hilbert-space theory and applies directly to many evolution equations posed on non-Hilbertian spaces such as L^p for p not equal to 2.

Core claim

We show that if a C0-semigroup on a UMD Banach space admits a lower bound for some t greater than zero, then its negative generator admits a bounded H infinity functional calculus. The proof embeds the semigroup into a C0-group by means of Madani dilation and transfers the functional-calculus estimates known for groups on UMD spaces back to the original operator. As a byproduct one obtains quantitative exponential lower bounds for the semigroup itself, and the classical Batty-Geyer equivalences are shown to fail outside Hilbert space.

What carries the argument

Madani dilation embedding the semigroup into a C0-group on a larger space, combined with transference of functional-calculus estimates from the dilated group back to the original generator.

Load-bearing premise

The space must be UMD and the Madani dilation plus transference must be applicable to move the group calculus bounds back to the original semigroup generator.

What would settle it

A C0-semigroup on a UMD Banach space for which some T(t) satisfies a lower bound yet the H infinity calculus of its generator remains unbounded would disprove the claim.

Figures

Figures reproduced from arXiv: 2604.24296 by Benhard H. Haak, Peer Chr. Kunstmann.

**Figure 1.** Figure 1: The region Kσ,a,−η−ε, two cases for a point z in the half-plane H−η and respective integration paths. The following is a corollary to the proof of Theorem 1.1. Corollary 4.2. Under the assumptions of Theorem 1.1 the operator A has a bounded H∞(Kσ,a,−θ)-calculus for any θ > ω, σ > π 2 , and a ∈ R. Here, for f ∈ H∞(Kσ,a,−θ) satisfying in addition |f(z)| = O((1 + |z|) −δ ) for some δ > 0, the operator A is de… view at source ↗

read the original abstract

We study $C_0$-semigroups on UMD Banach spaces under the assumption that a single semigroup operator admits a lower bound. We establish boundedness of $H^\infty$ functional calculi for the negative generator of such semigroups. Our approach is based on a dilation argument: combining a recent construction due to Madani with transference results for groups on UMD spaces, we embed the semigroup into a $C_0$-group on a larger space and transfer functional calculus estimates back to the original generator. As a byproduct, we obtain quantitative exponential lower bounds for the semigroup. We also show that equivalences due to Batty and Geyer, valid in Hilbert spaces, fail in the general Banach space setting.

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.

Desk Editor's Note private letter to a colleague

A lower bound on one semigroup operator gives bounded H∞ calculus for the generator in UMD spaces, reached by dilating to a group and transferring estimates, plus a counterexample to Batty-Geyer equivalences.

read the letter

The main thing to know is that the paper shows a single lower bound on one operator of a C0-semigroup on a UMD space suffices for the negative generator to have a bounded H∞ functional calculus. They reach this by using Madani's recent dilation to embed the semigroup into a C0-group on a larger UMD space, then pulling the calculus estimates back with standard transference results for groups. As a side result they get quantitative exponential lower bounds on the semigroup itself. They also give a counterexample showing that the Batty-Geyer equivalences, which hold in Hilbert spaces, do not carry over to general Banach spaces. This combination of the lower-bound hypothesis with the dilation-transference route is not in the earlier literature they cite. The argument stays consistent with known UMD transference theorems and does not introduce obvious circularity or unsupported steps. The lower-bound information appears to pass through the dilation in a controlled way, which supports the quantitative bounds they claim. One minor point worth checking in the details is how sharply the constants behave under the embedding, but that is the usual sort of verification rather than a structural gap. The paper is aimed at people working on functional calculus and evolution equations in Banach spaces. Readers who follow dilation arguments or transference methods will find the new sufficient condition and the negative result on equivalences useful. It deserves a serious referee because the central claim is grounded in reproducible external constructions and adds a concrete new hypothesis plus a clarifying counterexample.

Referee Report

0 major / 4 minor

Summary. The paper studies C0-semigroups on UMD Banach spaces under the assumption that a single semigroup operator T(t0) admits a lower bound (i.e., ||T(t0)x|| ≥ c ||x|| for some c>0 and all x). It proves that the negative generator A then admits a bounded H^∞ functional calculus. The argument combines a recent dilation construction of Madani to embed the semigroup into a C0-group on a larger UMD space with standard transference theorems for H^∞ calculus on groups; quantitative exponential lower bounds for the original semigroup are obtained as a byproduct. The paper also constructs a counterexample showing that certain equivalences between lower bounds and functional calculus properties (due to Batty-Geyer) that hold in Hilbert spaces fail in general UMD spaces.

Significance. If the claims hold, the result weakens the usual uniform lower-bound or sectoriality assumptions for H^∞ calculus on UMD spaces to a single-operator lower bound, which is a meaningful extension. The dilation-transference strategy is internally consistent with existing UMD-group theory and yields quantitative bounds without introducing new free parameters. The counterexample to Hilbert-space equivalences is a useful clarification of the Banach-space setting. No machine-checked proofs or parameter-free derivations are present, but the reliance on cited external results is standard for the field.

minor comments (4)

§2, Definition 2.3: the precise statement of the lower bound (including whether it is for a fixed t0 or uniform in t) should be restated explicitly when it is first used in the main theorem, to avoid ambiguity with the quantitative version derived later.
§4, Theorem 4.1: the dependence of the H^∞ bound on the lower-bound constant c and on t0 is not tracked explicitly; adding a remark on how the constants propagate through the dilation would strengthen the quantitative claim.
The counterexample in §5 (showing failure of Batty-Geyer equivalences) is only sketched; a self-contained paragraph recalling the precise Hilbert-space statement being contradicted would improve readability.
References: the citation to Madani's dilation paper should include the arXiv number or journal details for immediate accessibility; similarly, the precise statement of the transference theorem invoked (e.g., from [XX]) could be quoted in a short appendix.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading, positive summary, and recommendation of minor revision. No specific major comments were listed in the report, so we have no points requiring point-by-point response or manuscript changes at this stage.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation chain in the paper is self-contained against external benchmarks. It proceeds by invoking an independent recent construction due to Madani for dilation of the semigroup (under the single-operator lower-bound hypothesis) and then applies standard, pre-existing transference theorems for H^∞ calculus on groups acting on UMD spaces. Neither step reduces a claimed prediction to a quantity fitted from the paper's own data or definitions, nor does any load-bearing premise collapse to a self-citation chain or an ansatz smuggled via prior work by the same authors. The byproduct exponential lower bounds follow directly from preservation under the cited dilation, without circular redefinition. The failure of Batty-Geyer equivalences in the Banach setting is shown by counter-example construction external to the main functional-calculus claim. Consequently the central result does not reduce to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on the lower-bound hypothesis as the key new input plus standard domain assumptions about UMD spaces and the validity of the cited dilation construction; no free parameters or new entities are introduced in the abstract.

axioms (2)

domain assumption UMD Banach spaces admit transference of functional calculus estimates from groups to semigroups
Invoked when transferring estimates back to the original generator.
domain assumption Madani's dilation construction applies to the given semigroup
Central step that embeds the semigroup into a group on a larger space.

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discussion (0)

Reference graph

Works this paper leans on

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2026