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L∞-contraction for a weight fails on two points

Abstract

A damped rotation on two points is L∞-bounded but admits no contractive positive weight.

F. Madou, I. Remizov, M. Vafadar (eds.), Open problems in one-parameter operator semigroups theory, arXiv:2410.00416v2, §6 ‘L∞-bounds’, equation (1), 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), verbatim: “(3) ‖gT(t)f‖∞ ≤ ‖gf‖∞ for all f ∈ L∞(Ω) ∩ L²(Ω)” Problem 22 and its comment, 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.”

Definition 1.1 (The finite sup norm).

Formalization. D5/S3/Quantum/Semigroups/LinftyWeightContraction.supNorm (✓ std3).

Citation. F. Madou; I. Remizov; M. Vafadar (eds.) (2024). Open problems in one-parameter operator semigroups theory. DOI: 10.48550/arXiv.2410.00416. URL: https://arxiv.org/abs/2410.00416v2.

Commentary.

For u : Fin n → ℂ, supNorm u is the norm of the finite function, hence its sup norm. The definition includes the empty space n=0, where the norm is zero.

Definition 1.2 (Sectorial matrix forms).

Formalization. D5/S3/Quantum/Semigroups/LinftyWeightContraction.IsSectorialMatrix (✓ std3).

Citation. F. Madou; I. Remizov; M. Vafadar (eds.) (2024). Open problems in one-parameter operator semigroups theory. DOI: 10.48550/arXiv.2410.00416. URL: https://arxiv.org/abs/2410.00416v2.

Commentary.

For a complex n×n matrix A, q(A,u) denotes the finite sum of (A *ᵥ u) i times the complex conjugate of u i. IsSectorialMatrix A requires a real constant C with Re q(A,u)≥0 and |Im q(A,u)|≤C Re q(A,u) for every u. The associated form is a(u,v)=⟨Au,v⟩ on the full finite-dimensional space.

Definition 1.3 (The matrix exponential).

Formalization. D5/S3/Quantum/Semigroups/LinftyWeightContraction.semigroup (✓ std3).

Citation. F. Madou; I. Remizov; M. Vafadar (eds.) (2024). Open problems in one-parameter operator semigroups theory. DOI: 10.48550/arXiv.2410.00416. URL: https://arxiv.org/abs/2410.00416v2.

Commentary.

semigroup A t is NormedSpace.exp ((-t : ℂ) • A). The scalar ofReal(-t) is the real-to-complex embedding and smul is matrix scalar multiplication. For t≥0 this is the semigroup generated by −A.

Definition 1.4 (The finite-dimensional weight assertion).

Formalization. D5/S3/Quantum/Semigroups/LinftyWeightContraction.claim (✓ std3).

Source. Repository-derived.

Acknowledgement. F. Madou; I. Remizov; M. Vafadar (eds.) (2024). Open problems in one-parameter operator semigroups theory. DOI: 10.48550/arXiv.2410.00416. URL: https://arxiv.org/abs/2410.00416v2.

Commentary.

The claim ranges over every natural n and complex n×n matrix A. Sectoriality and a uniform real-time sup-norm bound imply the existence of a positive real weight g satisfying supNorm(g T(t)u)≤supNorm(g u) for every t≥0 and every complex vector u. weighted(g,u) means the function i↦(g i : ℂ)u i; mulVec is matrix action. The bound quantifies a real M without a separate M≥1 condition; the witness uses M=2. On Fin n with counting measure, L²∩L∞ is the full vector space, and a strictly positive weight and its inverse are bounded. Thus this is the finite-dimensional instance of Problem 22.

Theorem 1.5 (No positive weight contracts the damped rotation).

Proof. Machine-checked in Lean as D5/S3/Quantum/Semigroups/LinftyWeightContraction.result (✓ std3). ∎

Resolves. Problems/madou-remizov-vafadar-2024-linf-weight-contraction (refuted) by D5/S3/Quantum/Semigroups/LinftyWeightContraction.result.

Source. Repository-derived.

Acknowledgement. F. Madou; I. Remizov; M. Vafadar (eds.) (2024). Open problems in one-parameter operator semigroups theory. DOI: 10.48550/arXiv.2410.00416. URL: https://arxiv.org/abs/2410.00416v2.

Commentary.

Take A=[[1,2],[−2,1]]=I+2J with J=[[0,1],[−1,0]] and J²=−I. The lemma sectorial gives sector constant 2. The algebra map φ(a+bi)=aI+bJ, its continuity, and semigroup_formula give T(t)=e⁻ᵗ[[cos(2t),−sin(2t)],[sin(2t),cos(2t)]]. The lemmas entry_bound and bounded prove supNorm(T(t)u)≤2 supNorm(u) for t≥0. For any positive weight, no_weight tests (1/g₀,−1/g₁) in coordinate zero and (1/g₀,1/g₁) in coordinate one. Contractivity forces e⁻ᵗ(cos(2t)+c sin(2t))≤1 both for c=g₀/g₁ and for c=g₁/g₀. At least one coefficient exceeds 1/2. The lemma positive_bump uses the derivative 2c−1 at zero to produce a positive time with value greater than 1, contradicting that inequality. Hence result proves ¬ claim. In the source setting, use the two-point counting measure and V=ℂ². The form is closed and dense in finite dimension, with associated operator A. Since J*=−J, A*=I−2J and AA*=A*A=5I, so A is normal and unitarily diagonalizable. Its eigenvalues 1±2i satisfy Re(z(1±2i))≥|z|(cos θ−2 sin θ)≥0 on sectors |arg z|≤θ<arctan(1/2). Hence the L² operator norm of exp(−zA) is at most 1; the entire matrix exponential is bounded analytic there. These form and complex-time correspondences are mathematical reasoning; the module encodes the matrix sector bound and the real-time estimates. Together they give a negative answer to Problem 22. Geometrically, a diagonal weight rescales the square into a rectangle and conjugates the rotation generator. It cannot remove the outward corner motion: angular speed 2 and damping rate 1 would require both 2g₀/g₁≤1 and 2g₁/g₀≤1. Problems 21, 23, 24, and Problem 22 for positive semigroups are not settled by this result. Whether the finite-dimensional observation is known to specialists outside the problem list is ASSUMED-UNVERIFIED; no literature priority is established.

References