slug: madou-remizov-vafadar-2024-linf-weight-contraction
bibkey: madouremizovvafadar2024semigroups
doi: 10.48550/arXiv.2410.00416
url: https://arxiv.org/abs/2410.00416v2
triage: theorem
motivation_gids:
  - D5/S3/Quantum/Semigroups/LinftyWeightContraction.result

L∞-contraction for a weight

Problem

F. Madou, I. Remizov, M. Vafadar (eds.), Open problems in one-parameter operator semigroups theory, arXiv:2410.00416v2, §6 “L∞-bounds”, equation (1), 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 of Problem 21 is a σ-finite measure space $(\Omega ,\mu )$ and a closed sectorial sesquilinear form $a$ on a dense $V\subseteq L^2(\Omega )$, with associated operator $A$ and bounded analytic semigroup $T(t)$ generated by $-A$. The locator binds to the edited problem list, without assigning individual authorship to this question.

Motivation

A bounded positive weight with bounded inverse defines an equivalent weighted sup norm. The question asks whether the uniform bound $M$ can always be replaced by contractivity in a norm of this specific form.

Gap

The problem list presents Problem 22 as open. Its comment anticipates a negative answer but gives no counterexample. That source status does not establish the observation’s novelty outside the list.

Route

Use the two-point space with counting measure, $V={\mathbb{C}}^2$, and $a(u,v)=⟨Au,v⟩$, where

$$A=\left(\begin{array}{cc}1& 2\\ -2& 1\end{array}\right)=I+2J,J=\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right),J^2=-I.$$

The real part of $⟨Au,u⟩$ is $\Vert u{\Vert }_2^2$, and its imaginary part has absolute value at most $2\Vert u{\Vert }_2^2$. Thus the matrix satisfies IsSectorialMatrix with constant 2. The algebra map $\phi (a+bi)=aI+bJ$ transports the scalar exponential to the matrix exponential, giving

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

Each entry has modulus at most 1 for $t\ge 0$, so $\Vert T(t)u{\Vert }_{\infty }\le 2\Vert u{\Vert }_{\infty }$. The finite-dimensional form is closed on the full, dense space, and its associated operator is $A$. For the $L^2$ step, $J^{\ast }=-J$, hence $A^{\ast }=I-2J$ and $A{A}^{\ast }=A^{\ast }A=5I$. Thus $A$ is normal and unitarily diagonalizable, with eigenvalues $1\pm 2i$. For $z$ in a sector $|\arg z|\le \theta <\arctan (1/2)$,

$$\mathrm{Re}(z(1\pm 2i))\ge |z|(\cos \theta -2\sin \theta )\ge 0,|e^{-z(1\pm 2i)}|\le 1.$$

Unitary diagonalization therefore gives $\Vert e^{-zA}{\Vert }_{2\to 2}\le 1$ throughout that sector. The matrix exponential is entire, so this proves sector contractivity and bounded analyticity on $L^2$. These form and complex-time identifications explain the source correspondence; the Lean predicates encode sectoriality and the real-time matrix bound.

Falsifier

Let $g_0,g_1>0$ and $r=g_0/g_1$. The real vectors $(1/g_0,-1/g_1)$ and $(1/g_0,1/g_1)$ both have weighted sup norm 1. Testing the zeroth coordinate of the first image and the first coordinate of the second image forces

$$e^{-t}(\cos (2t)+r\sin (2t))\le 1,e^{-t}(\cos (2t)+r^{-1}\sin (2t))\le 1(t\ge 0).$$

For $f_c(t)=e^{-t}(\cos (2t)+c\sin (2t))$, one has $f_c(0)=1$ and $f_c^{\prime }(0)=2c-1$. If $c>1/2$, then $f_c(t)>1$ at some positive time. Either $r>1/2$, or $r\le 1/2$ and $r^{-1}>1/2$. One of the two required inequalities therefore fails. Equivalently, their right derivatives would require both $2r\le 1$ and $2/r\le 1$.

Evidence

The public declarations in D5/S3/Quantum/Semigroups/LinftyWeightContraction.lean are supNorm, IsSectorialMatrix, semigroup, claim, and result. The witness matrices A, J, the algebra map φ, and all supporting lemmas are private.

supNorm u is the norm of the finite function $u:\mathrm{Fin}n\to \mathbb{C}$, including $n=0$. IsSectorialMatrix A asserts the existence of a real $C$ such that every $u$ has nonnegative real quadratic form and $|\mathrm{Im}⟨Au,u⟩|\le C\mathrm{Re}⟨Au,u⟩$. semigroup A t is NormedSpace.exp ((-t : ℂ) • A).

claim quantifies over every natural $n$ and every complex $n\times n$ matrix $A$. Sectoriality and the existence of a real uniform bound $M$ imply the existence of $g:\mathrm{Fin}n\to \mathbb{R}$, positive at every coordinate, such that $\Vert gT(t)u{\Vert }_{\infty }\le \Vert gu{\Vert }_{\infty }$ for every $t\ge 0$ and every complex vector $u$. The formal bound quantifies $M:\mathbb{R}$ without a separate $M\ge 1$ condition; the counterexample uses $M=2$, so it also meets the source condition.

sectorial checks the sector bound. J_sq, φ, φ_apply, and φ_continuous establish the algebra-map calculation used in semigroup_formula. entry_bound and bounded give the bound 2. positive_bump obtains a positive-time violation from the derivative at zero. action gives the coordinate formula, and no_weight tests the two weighted vectors. The theorem D5.S3.Quantum.Semigroups.LinftyWeightContraction.result has type ¬ claim.

The module’s axiom closure is $\{\mathrm{propext},Classical.choice,Quot.sound\}$, with no sorry or additional axiom. On the two-point space $L^2\cap L^{\infty }={\mathbb{C}}^2$, and every strictly positive weight and its inverse are bounded. The finite-dimensional correspondence described in Route places the matrix counterexample in the source’s sectorial-form setting.

Triage

What the settlement shows

1. Proved (Lean) — the counterexample. The finite-dimensional matrix instance is sectorial, L∞-bounded, and admits no contractive positive weight. Via its finite-dimensional sectorial-form interpretation, the answer to Problem 22 is negative.

2. Computed (exact mathematical reasoning) — the mechanism. A diagonal weight conjugates the rotation generator but cannot eliminate rotation. The real slice of the ℓ∞ unit ball is a square whose corners rotation pushes outward. Diagonal rescaling turns it into a rectangle; with angular speed 2 and damping rate 1, at least one corner still moves outward at small positive times, since contractivity would require both $2r\le 1$ and $2/r\le 1$. A scalar diagonal weight commutes with rotation, but does not remove this obstruction; a nonscalar diagonal weight does not commute with it.

3. Open — Problem 21. Characterization of L∞-boundedness via the form is not settled by this result.

4. Open — Problem 23. Characterization of weighted L∞-contractive semigroups is not settled by this result.

5. Open — Problem 24. Construction of weights is not settled by this result.

6. Open — positive semigroups. Problem 22 under the positivity assumption discussed in the comments is not settled by this result. The counterexample’s real-time matrix has a negative off-diagonal entry for small positive times, so it does not preserve the positive cone.

ASSUMED-UNVERIFIED

Whether the finite-dimensional observation is known to specialists outside the problem list is unverified. The quotations bind to arXiv:2410.00416v2, §6, equations (1), (3), and Problem 22; no literature priority is established.

Closedness and density of the finite-dimensional form, identification of its associated operator, and bounded analyticity are mathematical correspondences described in Route, not separate statements of the Lean module. The matrix sector bound, real-time uniform bound, and failure of every positive weight are the formal conclusions used in the refutation.