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Cone Minimax Principles for Non-Selfadjoint Operator Pencils

This paper proposes a variational approach using extended two-variable Rayleigh quotients on admissible cone pairs to establish Rayleigh-type minimax principles for characterizing selected real spectral values of non-selfadjoint operator pencils with singular weights, thereby extending classical finite-dimensional approximation methods and providing stability guarantees, error estimates, and spectral certificates in settings where standard selfadjoint theory is unavailable.

Original authors: Yavdat Il'yasov, Nur Valeev

Published 2026-07-01

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CC BY 4.0

Original authors: Yavdat Il'yasov, Nur Valeev

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to find the "heartbeat" or the most important rhythm of a complex machine. In the world of mathematics, this machine is a system of equations called an operator pencil. Usually, if the machine is perfectly symmetrical (what mathematicians call "self-adjoint"), finding this rhythm is easy. You have a well-known map called the "Rayleigh quotient" that acts like a GPS, guiding you straight to the answer.

However, this paper deals with machines that are not symmetrical (non-selfadjoint). These are messy, lopsided systems where the standard GPS breaks down. Furthermore, the "weight" of the machine (the part that tells us how heavy or important different parts are) might be broken or missing in places (singular). In these messy scenarios, traditional methods often fail because they rely on the machine being perfectly positive or symmetrical.

Here is what the authors, Yavdat Il'yasov and Nur Valeeva, have invented to solve this problem:

1. The Two-Variable "Tug-of-War"

Instead of looking at the machine from just one angle, the authors propose looking at it from two different angles at once.

Imagine a game of tug-of-war.

Player A (u) is trying to pull the rope in a direction that makes the machine's "energy" look as low as possible.
Player B (v) is trying to pull in a direction that makes the energy look as high as possible.

Both players are restricted to a specific "cone" of allowed moves. Think of this cone as a rulebook that says, "You can only pull in positive directions." The authors define a special scorecard (an extended Rayleigh quotient) that measures the tension between these two players.

2. The "Minimax" Strategy

The goal is to find a sweet spot where the game stabilizes.

The "Minimax" Level: Player A tries to find the best move to minimize the score, knowing Player B will try to maximize it.
The "Maximin" Level: Player B tries to find the best move to maximize the score, knowing Player A will try to minimize it.

In a perfect, symmetrical world, these two levels would meet in the middle. In this messy, non-symmetrical world, they usually don't. But the authors prove that if you find a specific pair of moves where these two levels collide (meet at the same number), you have found the machine's true "principal spectral value"—its most important rhythm.

3. The "Trap" and the "Saddle"

The paper introduces a clever concept called a "cone quasi-eigenvalue."

Think of a landscape with hills and valleys. Usually, you look for the highest peak or the deepest valley.
In this non-symmetrical world, the "true" value might not be a peak or a valley at all. It might be a saddle point (like the spot between two hills where you can go up or down depending on which way you face).
The authors show that even if the machine doesn't have a standard "eigenvalue" (a clean solution), this saddle point still exists and acts as a real, measurable value for the system. It's like finding a stable balance point on a wobbly table.

4. The "Certificate" (The Best Part)

One of the most practical features of this method is that it acts like a certificate of accuracy.

In many math problems, you calculate an answer and hope it's right.
Here, the method gives you a lower bound (a floor) and an upper bound (a ceiling) simultaneously.
Imagine you are guessing the temperature. Instead of just saying "It's 70 degrees," this method says, "We know for a fact it is at least 68 and at most 72."
As you refine your calculation (using computer approximations), this gap between the floor and ceiling gets smaller and smaller, giving you a guaranteed range for the answer. You don't need to solve the whole puzzle perfectly to know the answer is within a specific, safe range.

5. Why This Matters

The authors tested this on complex systems, including:

Systems where the "weight" is broken (singular), meaning standard math tools can't even start.
Systems that are "non-cooperative," where the parts of the machine fight against each other rather than helping.

They proved that even in these difficult cases, you can use this two-player tug-of-war method to find the machine's core rhythm. They also showed that if you break the problem down into small, manageable chunks (finite-dimensional approximation), the answer you get from the chunks will eventually converge to the true answer of the whole machine.

Summary Analogy

Imagine trying to find the center of gravity of a weirdly shaped, lopsided sculpture that is floating in a foggy room.

Old Method: You need a perfect, symmetrical sculpture to use a laser level. If it's lopsided, the laser fails.
This Paper's Method: You send two people into the fog. One tries to find the lowest point, the other the highest. They communicate back and forth. Even if the sculpture is weird, they eventually find a spot where their reports match up. That matching spot is the true center of gravity. Furthermore, at every step, they can tell you, "We know the center is definitely between these two markers," giving you a safe, verified range without needing to see the whole sculpture clearly.

The paper essentially provides a new, robust way to find the "heart" of messy, asymmetrical mathematical systems, complete with a built-in safety check to ensure the answer is reliable.

Technical Summary: Cone Minimax Principles for Non-Selfadjoint Operator Pencils

Problem Statement
The paper addresses the spectral analysis of non-selfadjoint operator pencils of the form $Lu = \lambda Gu$ , where $L$ is a bounded linear operator (not necessarily selfadjoint or invertible) and $G$ is a weight operator that may be singular. Standard variational theories, such as the Courant–Fischer principle and Rayleigh quotient minimization, are generally unavailable for non-selfadjoint operators. While positivity-based methods (e.g., Perron–Frobenius, Krein–Rutman) exist, they typically require the operator to be selfadjoint, the weight to be invertible, or the resolvent to be positive. The authors seek a variational characterization for selected real spectral levels in settings where these standard assumptions fail, specifically when $G$ is singular and the operator lacks a positive resolvent structure.

Methodology
The core methodology introduces an extended two-variable Rayleigh quotient defined on admissible pairs of vectors within an ordered cone $S$ and its strictly positive part $S_o$ :
$R(u, v) = \frac{\langle Lu, v \rangle}{(Gu, v)_H}$
where $u$ and $v$ are independent cone variables. The denominator is assumed positive on mixed cone pairs.

The authors define two principal minimax levels:

Right sup–inf level: $\lambda = \sup_{u \in S \setminus \{0\}} \inf_{v \in S_o} R(u, v)$
Left inf–sup level: $\lambda = \inf_{v \in S \setminus \{0\}} \sup_{u \in S_o} R(u, v)$

The framework relies on the concept of cone quasi-eigenvalues and quasi-eigenvectors, defined not by algebraic eigenvalue equations but by the realization of these minimax profiles. The theory is developed in three stages:

Abstract Framework: Establishing the relationship between the minimax levels and quasi-eigenvalues in a general Banach-Hilbert space setting.
Discrete Approximation: Applying the framework to finite-dimensional Galerkin approximations (specifically conforming finite elements), proving that under strict positivity and boundary-exclusion conditions, the discrete minimax levels collapse to exact positive right-left eigenpairs.
Continuous Limit: Passing to the infinite-dimensional limit using compactness arguments and strong positivity conditions (cone formulations of the strong maximum principle) to identify the continuous minimax value with the spectral value of the pencil.

Key Contributions and Results

Variational Characterization: The paper establishes that if a strictly positive right-left eigenpair exists, the associated eigenvalue coincides with the common value of the minimax levels ( $\hat{\lambda} = \lambda = \lambda$ ). Crucially, the method yields a real "cone level" even when the operator has no real eigenvalues (e.g., in cases of non-real spectra where the skew-symmetric part is "invisible" to the cone test).
Trapping and Saddle-Point Principles: The authors prove that every cone quasi-eigenvalue is trapped between the two minimax levels ( $\lambda \leq \lambda^* \leq \lambda$ ). If the realizing vectors are strictly positive, the levels collapse, and the common value admits a saddle-point interpretation.
A Posteriori Spectral Certificates: A central result is the provision of computable one-sided bounds. For any admissible trial vectors $u$ and $v$ , the quantities $\lambda_R(u) = \inf_{v} R(u, v)$ and $\lambda_L(v) = \sup_{u} R(u, v)$ satisfy $\lambda_R(u) \leq \hat{\lambda} \leq \lambda_L(v)$ . This allows for direct verification of bounds without solving the full eigenvalue problem.
Finite-Dimensional Stability: The paper proves that the continuous minimax value is the limit of discrete minimax values obtained via standard finite element approximations. The discrete levels provide exact eigenpairs for the discretized pencil, and their convergence to the continuous limit is guaranteed under uniform ellipticity and strong positivity assumptions.
Perturbation Estimates: The authors derive one-sided perturbation bounds. Unlike classical analytic perturbation theory which relies on derivatives of isolated eigenvalues, this method bounds the variation of the spectral level using one-sided quantities derived from the quasi-eigenvectors. These estimates are shown to be sharp in model examples.
Handling Singular Weights: The method successfully treats cases where $G$ is singular (non-invertible), a scenario where shifted resolvent approaches (Krein–Rutman) often fail or require auxiliary shifts.

Significance and Claims
The authors claim that this work provides a variational and computational tool for selected real spectral values of non-selfadjoint operator pencils, particularly in ordered cone settings where standard selfadjoint theory is inapplicable and positivity-based methods are not directly usable.

Key aspects of the paper's significance include:

Independence from Resolvent Positivity: The method treats the quotient $R(u, v)$ as the primary object, avoiding the need to construct a positive resolvent or shift the operator to ensure invertibility.
Practical Computability: The framework yields a posteriori spectral certificates. The gap between the lower and upper bounds (the primal-dual gap) serves as a practical stopping criterion for numerical approximations.
Robustness: The theory accommodates singular weights and non-cooperative systems (where the standard componentwise positive cone is not invariant), extending the scope of variational spectral analysis beyond traditional cooperative or selfadjoint cases.
Unification: It unifies the classical Collatz–Wielandt and Birkhoff–Varga formulas for matrices with the spectral theory of elliptic operator pencils, offering a consistent approach for both finite and infinite-dimensional problems.

The paper concludes that the cone minimax approach offers a robust alternative for spectral analysis, providing not just existence results but also direct, verifiable bounds and approximation estimates for non-selfadjoint systems.