bibkey: kushelzabreiko2008ideal authors: Olga Y. Kushel and Petr P. Zabreiko year: 2008 title: Gantmakher–Krein theorem for 2-totally nonnegative operators in ideal spaces doi: null url: https://arxiv.org/abs/0812.0902v1 claim: The source requires positivity preservation on an ideal function space, together with exterior-square positivity and further operator hypotheses. The inherited pointwise cone on the original theta remainder is trivial, so its nonzero compressed semigroup and resolvent cannot meet this interface in the original radial coordinates. strata_touched: [] license: bibliographic-reference-only triage: anchor

Exterior-square positivity and the original theta remainder

Published hypotheses

The inspected primary preprint is arXiv:0812.0902v1. Its PDF has SHA-256 10cc9d4d2207731b83c71ff3514ce0b5197304f0c8ccb07629968eb95b521c28. Only bibliographic information and the following source/application distinctions are retained; no third-party PDF or implementation is copied into the project.

Section2, PDF p.3, defines a Banach ideal function space by solidity: if $|f|\le |g|$ almost everywhere and $g$ belongs to the space, then so does $f$, with the corresponding norm bound. Its nonnegative cone is the cone of almost-everywhere nonnegative functions. Section4, PDF p.6, defines regular operators as differences of operators preserving this cone, and resolvent-regularity requires regularity of every resolvent outside the spectrum.

Section7, Theorem3, PDF p.10, assumes an almost perfect ideal space, a compact, positivity-preserving, resolvent-regular operator with positive spectral radius, and a positivity-preserving exterior square with positive spectral radius. It obtains a leading positive eigenvalue and a second-eigenvalue alternative subject to the stated spectral-circle conditions. Simplicity and nodal ordering are not borrowed from this statement without their additional hypotheses. Theorem4, PDF pp.10–11, gives the integral-kernel version: both the original kernel and its second associated kernel must be nonnegative almost everywhere and not identically zero on their stated domains. Neither theorem supplies a prescribed numerical bound such as one-half.

The remark on PDF p.11 allows an arbitrary almost reproducing cone for the exterior square. It does not remove the original operator’s positivity hypothesis or provide a transported cone for a given orthogonal remainder. Quadratic nonnegativity in a Hilbert space is a different condition from preserving a cone of nonnegative functions.

The inherited cone is lost after critical projection

Use the original minimal even theta form with probability measure $d\nu =2\Phi (x)\cosh (x/2)dx$, nonnegative self-adjoint energy operator $A$, and $A1=0$. Retain the critical decomposition into constants, the closed span $N$ of the known real one-half eigenvectors, and the nontrivial reducing remainder

$$\mathcal{R}=(\mathbb{C}1\oplus N{)}^{\perp },Q=Q_{\mathcal{R}}.$$

All these operator/model premises are the existing paper-level source applications. The following interface check is conditional on them; it is not a Lean-certified operator construction.

Since $\nu$ is finite, every $r\in \mathcal{R}$ is integrable and has $\nu (r)=0$. For a real $r\ge 0$ almost everywhere, the usual nonnegative-integral criterion gives $r=0$ almost everywhere. Thus

$$\mathcal{R}\cap L_{+}^2(\nu )=\{0\}.\text{\hspace{1em}(1)}$$

The original measure and radial order are retained here. In particular, $\mathcal{R}$ is not an ideal function space with that order: for any nonzero $r\in \mathcal{R}$, the function $|r|$ belongs to ambient $L^2$ and has positive mean, so $|r|\notin \mathcal{R}$. Solidity would require it to belong to $\mathcal{R}$. Restricting positivity preservation to the zero cone in (1) would be vacuous and would not supply the source’s ideal-space hypotheses.

There is a stronger ambient obstruction. Suppose a complex-linear operator $B$ on the original even $L^2(\nu )$ preserves nonnegative real functions and has range in $\mathcal{R}$. Equation(1) gives $Bf=0$ for every nonnegative $f$. Positive and negative parts span the real even space, and complexification spans the whole even space, so $B=0$.

For $t>0$ and $\alpha >0$, consider the bounded compressions

$$K_t=Q{e}^{-tA}Q,G_{\alpha }=Q(A+\alpha {)}^{-1}Q.\text{\hspace{1em}(2)}$$

They have range in $\mathcal{R}$ and are nonzero. Indeed, for any nonzero $r\in \mathcal{R}$, the spectral measure ${\mu }_r$ of $A$ gives

$$⟨r,K_{t}r⟩={\int }_{[0,\infty )}{e}^{-ts}d{\mu }_r(s)>0,⟨r,G_{\alpha }r⟩={\int }_{[0,\infty )}\frac{d{\mu }_r(s)}{s+\alpha }>0.\text{\hspace{1em}(3)}$$

Consequently neither compression preserves the ambient pointwise nonnegative cone. In any ordinary integral-kernel representation acting on that entire ambient even/radial space, its kernel cannot be nonnegative almost everywhere: such a kernel would preserve this cone. No existence or regularity of such a kernel is assumed. This rules out the direct application of the source’s nonnegative-kernel criterion in the original coordinates before any ordered-minor calculation.

This conclusion is separate from the full semigroup’s ordered-jump obstruction. It requires no compact support in $\mathcal{R}$, no projected bump calculation, and no compactness assertion about $A{|}_{\mathcal{R}}$. The known critical vectors lie in $N$, not in $\mathcal{R}$, so they cannot be identified as the first or second eigenvectors of an operator whose domain is $\mathcal{R}$.

The quantitative obligation remains

The cone obstruction does not decide the desired lower bound on $\mathcal{R}$. A different ordered realization would require an explicit map from its function space and cone to this same remainder, verification of the published operator hypotheses, and a spectral identification that produces the numerical threshold. Merely naming a new cone or observing quadratic nonnegativity of an exterior power does not provide that map or estimate.

For the already specified nonnegative self-adjoint restriction $A_{\mathcal{R}}$, the spectral theorem gives the standard equivalence

$$D(r)\ge \frac{1}{2}\Vert r{\Vert }_{\nu }^2(r\in \mathcal{R}\cap \mathcal{F})⟺\Vert (A_{\mathcal{R}}+\alpha {)}^{-1}\Vert \le \frac{1}{\alpha +1/2},\text{\hspace{1em}(4)}$$

for any fixed $\alpha >0$. Equation(4) is a reformulation, not a new resolvent estimate. The norm bound remains unproved; this source check does not settle RH, Robin, or the original cofinal signed comparison.