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arxiv: 2505.05788 · v3 · submitted 2025-05-09 · 🧮 math.FA

H^infty Functional Calculus for a Commuting Pair of Ritt_(E) Operators

Suman Mondal , Subhajit Palai , Samya Kumar Ray This is my paper

Pith reviewed 2026-05-22 17:06 UTC · model grok-4.3

classification 🧮 math.FA
keywords Ritt_E operatorsfunctional calculuscommuting operatorsBanach spacesL^p spacessectorial operatorsdilation theoremtransfer principle
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The pith

Commuting pairs of Ritt_E operators on Banach spaces admit a joint bounded holomorphic functional calculus precisely when equivalent criteria on L^p spaces hold.

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

The paper builds a framework for the joint functional calculus of commuting Ritt_E operator pairs on Banach spaces. It establishes a transfer principle that links the bounded holomorphic calculus of these operators to the calculus of their associated sectorial operators. The authors also prove a joint dilation theorem that applies to commuting tuples of Ritt_E operators on a wide range of Banach spaces. The central payoff is an explicit list of equivalent conditions on L^p spaces for 1 < p < ∞ that determine exactly when the pair possesses the joint bounded calculus.

Core claim

The paper establishes that for commuting pairs of Ritt_E operators, the existence of a joint bounded H^∞ functional calculus is equivalent to a set of conditions that can be verified by reducing to the case of sectorial operators via a transfer principle and using a joint dilation theorem on appropriate Banach spaces, with explicit criteria provided for L^p spaces.

What carries the argument

The transfer principle relating the bounded holomorphic functional calculus for Ritt_E pairs to their sectorial counterparts, combined with the joint dilation theorem for commuting tuples.

If this is right

  • The joint calculus problem for Ritt_E pairs reduces directly to the corresponding sectorial operator problem.
  • The dilation theorem extends the result to many Banach spaces beyond Hilbert spaces, including L^p spaces.
  • Verification of the joint bounded calculus reduces to checking a finite list of equivalent conditions formulated on L^p spaces.
  • When the criteria hold, the pair generates a bounded joint holomorphic functional calculus on the appropriate function algebra.

Where Pith is reading between the lines

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

  • The same transfer and dilation approach may apply to commuting tuples larger than pairs.
  • The criteria could be used to study stability and spectrum properties of the generated semigroup in evolution equations.
  • Analogous reduction techniques might work for other operator classes that admit sectorial-like dilations on Banach spaces.

Load-bearing premise

The transfer principle and joint dilation theorem hold for commuting pairs of Ritt_E operators on a broad class of Banach spaces, allowing reduction to the sectorial case.

What would settle it

A concrete commuting pair of Ritt_E operators on an L^p space for which the listed equivalent criteria fail yet a joint bounded H^∞ calculus still exists, or vice versa.

Figures

Figures reproduced from arXiv: 2505.05788 by Samya Kumar Ray, Subhajit Palai, Suman Mondal.

Figure 1
Figure 1. Figure 1: polygon ∆0 i and ∆i [PITH_FULL_IMAGE:figures/full_fig_p014_1.png] view at source ↗
read the original abstract

In this article, we develop a framework for the joint functional calculus of commuting pair of $\text{Ritt}_{\text{E}}$ operators on Banach spaces. We establish a transfer principle that relates the bounded holomorphic functional calculus for pair of $\text{Ritt}_{\text{E}}$ operators to that of their associated sectorial counterparts. In addition, we prove a joint dilation theorem for commuting tuples of $\text{Ritt}_{\text{E}}$ operators on a broad class of Banach spaces. As a key application, we obtain an equivalent set of criteria on $L^p$-spaces for $1<p< \infty$ that determine when a commuting pair of $\text{Ritt}_{\text{E}}$ operators admits a joint bounded functional calculus.

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

0 major / 2 minor

Summary. The manuscript develops a framework for the joint H^∞ functional calculus of commuting pairs of Ritt_E operators on Banach spaces. It establishes a transfer principle relating the bounded holomorphic functional calculus for such pairs to their sectorial counterparts and proves a joint dilation theorem for commuting tuples of Ritt_E operators on a broad class of Banach spaces. As a key application, it derives an equivalent set of criteria on L^p-spaces (1 < p < ∞) determining when a commuting pair admits a joint bounded functional calculus.

Significance. If the central results hold, the work is a solid contribution to operator theory on Banach spaces. It extends single-operator results on Ritt_E and sectorial operators to the commuting-pair setting via a transfer principle and joint dilation, providing a reduction that yields concrete equivalent criteria on L^p spaces. This is useful for applications and builds directly on established tools without introducing free parameters or ad-hoc constructions.

minor comments (2)
  1. The abstract and introduction would benefit from a brief explicit statement of the precise class of Banach spaces on which the joint dilation theorem holds, to make the scope immediately clear to readers.
  2. Notation for the joint functional calculus (e.g., the precise definition of the joint H^∞ calculus for pairs) should be introduced with a numbered equation in the preliminaries section for easier cross-reference.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript on the joint H^∞ functional calculus for commuting pairs of Ritt_E operators. We appreciate the recommendation for minor revision and will prepare a revised version accordingly. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The manuscript establishes a transfer principle relating the joint bounded holomorphic functional calculus for commuting Ritt_E operator pairs to their sectorial counterparts, together with a joint dilation theorem on a broad class of Banach spaces. These constructions are then applied to obtain equivalent criteria on L^p spaces (1 < p < ∞) by reducing the Ritt_E case to the sectorial setting and invoking known single-operator results in the commuting case. No step reduces by definition or construction to its own inputs, no parameters are fitted and relabeled as predictions, and no load-bearing premise rests on self-citation chains; the argument is self-contained against external operator-theoretic benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities identifiable from abstract alone; full paper would be needed to audit assumptions such as spectral conditions or space properties.

pith-pipeline@v0.9.0 · 5660 in / 965 out tokens · 61375 ms · 2026-05-22T17:06:11.249347+00:00 · methodology

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Reference graph

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