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bibkey: madouremizovvafadar2024semigroups
authors: F. Madou; I. Remizov; M. Vafadar (eds.)
year: 2024
title: "Open problems in one-parameter operator semigroups theory"
doi: 10.48550/arXiv.2410.00416
url: https://arxiv.org/abs/2410.00416v2
claim: "Section 6, Problem 22 asks whether every L∞-bounded analytic semigroup in the sectorial-form setting admits a bounded strictly positive weight with bounded inverse making it L∞-contractive."
strata_touched:
  - D5/S3/Quantum/Semigroups/LinftyWeightContraction
  - D5/S3/Quantum/Semigroups/InteractionPictureAsymptotics
license: citation-only
triage: anchor

Locator

Author text: https://arxiv.org/abs/2410.00416v2.

• DOI: https://doi.org/10.48550/arXiv.2410.00416 F. Madou, I. Remizov, M. Vafadar (eds.), Open problems in one-parameter operator semigroups theory, arXiv:2410.00416v2, §6 “L∞-bounds”, Problems 21–24, specifically Problem 22 and equations (1), (3). The same list states Problem 29 (A. E. Teretenkov) on p. 21, on interaction-picture asymptotics for perturbed semigroups.

The weight question

Equation (1) and its defining sentence state verbatim:

The semigroup (T(t))t≥0 is called L∞-bounded, if there exists M > 0 such that ‖T(t)f‖∞ ≤ M‖f‖∞, for all f ∈ L²(Ω) ∩ L∞(Ω). (1)

Equation (3) states verbatim:

(3) ‖gT(t)f‖∞ ≤ ‖gf‖∞ for all f ∈ L∞(Ω) ∩ L²(Ω)

Problem 22 and its comment state verbatim:

Problem 22 (L∞-contraction for a weight problem). In the setting of Problem 21 assume that the semigroup (T(t))t≥0 is L∞-bounded, i.e., satisfies (1) for some M ≥ 1. Does there exist a function g ∈ L∞(Ω) satisfying g > 0 µ-a.e., 1/g ∈ L∞(Ω), and (3)? Comments. In general, this might be more than one can hope for.

The setting is a σ-finite measure space, a closed sectorial sesquilinear form on a dense subspace of L², its associated operator A, and the bounded analytic semigroup generated by −A. The volume is an edited problem list; the weight question is attributed here to its §6, without assigning individual authorship.

Repository-derived result

The theorem D5.S3.Quantum.Semigroups.LinftyWeightContraction.result : ¬ claim gives a two-point counterexample. With counting measure, take V = ℂ² and the form a(u,v) = ⟨Au,v⟩ for

$$A=\left(\begin{array}{cc}1& 2\\ -2& 1\end{array}\right),T(t)=e^{-t}\left(\begin{array}{cc}\cos (2t)& -\sin (2t)\\ \sin (2t)& \cos (2t)\end{array}\right).$$

The matrix form is sectorial with constant 2, and the semigroup has sup-norm bound 2 for t ≥ 0. A positive weight has ratio r = g₀/g₁. Contractivity would require both e⁻ᵗ(cos(2t) + r sin(2t)) ≤ 1 and e⁻ᵗ(cos(2t) + r⁻¹ sin(2t)) ≤ 1. Their right derivatives at zero require 2r ≤ 1 and 2/r ≤ 1, which are incompatible. The Lean proof obtains a strictly positive time violating one inequality from its positive derivative at zero.

Writing A = I + 2J, one has J* = −J and AA* = A*A = 5I, so A is normal and unitarily diagonalizable. For |arg z| ≤ θ < arctan(1/2), its eigenvalues satisfy Re(z(1±2i)) ≥ |z|(cos θ − 2 sin θ) ≥ 0. Thus ‖exp(−zA)‖₂→₂ ≤ 1, and the entire matrix exponential is bounded analytic on that sector.

On a finite space every strictly positive weight and its inverse are bounded. Closedness, density, identification of the associated operator, and bounded analyticity are the finite-dimensional correspondence with the source setting, rather than separate predicates of the matrix module.

ASSUMED-UNVERIFIED

Whether this finite-dimensional observation is known to specialists outside the problem list is unverified. The result establishes no literature priority. Problems 21, 23, 24, and Problem 22 with the positivity assumption discussed in the comments are not settled by this result.