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A Scalar Analytic Characterization for Dominant Spectral Poles under Rank-One Minorization

Yuki Chino, Kensaku Kinjo, Ryo Oizumi

TL;DR

This work develops a determinant-free, resolvent-based framework to characterize the dominant spectral pole of a positive operator on a Banach lattice under a rank-one Doeblin-type minorization. By decomposing the operator as $T=\alpha P_0+R$ and employing a Birman–Schwinger resolvent factorization, the isolated spectrum is governed by a scalar analytic function $D(\lambda)=1-\alpha\Phi[R_\lambda u_0]$, with the dominant eigenvalue given by the unique zero of $D$ and the corresponding spectral projection realized as a rank-one residue. The authors also construct a kernel-level resolvent expansion via corrected kernels $\Gamma_n$, enabling explicit, constructive expressions for resolvent kernels and eigenfunctions. The results extend classical Perron–Frobenius/Krein–Rutman theory to non-compact Banach lattices by leveraging Doeblin-type minorization to achieve quasi-compactness and positivity-improvement, and they reveal a clear Euler–Lotka-type interpretation of the dominant growth rate. The framework provides a versatile template for analyzing integral projection models and related spectral problems without requiring Fredholm determinants or trace-class assumptions.

Abstract

This paper provides a resolvent-based, determinant-free characterization of the dominant spectral pole for positive operators on Banach lattices under a rank-one Doeblin-type minorization. Departing from traditional requirements of compactness or trace-class properties, we demonstrate that the dominant eigenvalue is strictly positive, algebraically simple, and uniquely identified as the zero of a Birman--Schwinger-type scalar analytic function. The associated spectral projection is explicitly obtained as a rank-one residue. Our approach reduces complex spectral problems to the analysis of a scalar function, providing a bridge between abstract Krein--Rutman theory and constructive operator methods.

A Scalar Analytic Characterization for Dominant Spectral Poles under Rank-One Minorization

TL;DR

This work develops a determinant-free, resolvent-based framework to characterize the dominant spectral pole of a positive operator on a Banach lattice under a rank-one Doeblin-type minorization. By decomposing the operator as and employing a Birman–Schwinger resolvent factorization, the isolated spectrum is governed by a scalar analytic function , with the dominant eigenvalue given by the unique zero of and the corresponding spectral projection realized as a rank-one residue. The authors also construct a kernel-level resolvent expansion via corrected kernels , enabling explicit, constructive expressions for resolvent kernels and eigenfunctions. The results extend classical Perron–Frobenius/Krein–Rutman theory to non-compact Banach lattices by leveraging Doeblin-type minorization to achieve quasi-compactness and positivity-improvement, and they reveal a clear Euler–Lotka-type interpretation of the dominant growth rate. The framework provides a versatile template for analyzing integral projection models and related spectral problems without requiring Fredholm determinants or trace-class assumptions.

Abstract

This paper provides a resolvent-based, determinant-free characterization of the dominant spectral pole for positive operators on Banach lattices under a rank-one Doeblin-type minorization. Departing from traditional requirements of compactness or trace-class properties, we demonstrate that the dominant eigenvalue is strictly positive, algebraically simple, and uniquely identified as the zero of a Birman--Schwinger-type scalar analytic function. The associated spectral projection is explicitly obtained as a rank-one residue. Our approach reduces complex spectral problems to the analysis of a scalar function, providing a bridge between abstract Krein--Rutman theory and constructive operator methods.
Paper Structure (34 sections, 13 theorems, 112 equations)

This paper contains 34 sections, 13 theorems, 112 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Structural foundations
  4. Discussion
  5. Organization of the paper
  6. Framework and Structural Properties of the Operator
  7. Banach lattices and order continuity
  8. Doeblin-type minorization
  9. Positivity-improving property and irreducibility
  10. Rank-one decomposition and quasi-compactness
  11. Rank-One Perturbation and the Resolvent Factorization
  12. Inversion formula for rank-one perturbations
  13. Birman--Schwinger resolvent factorization of the resolvent of
  14. Simple poles and rank-one spectral projections
  15. Remarks on the function
  16. ...and 19 more sections

Key Result

Lemma 2.2

Assume Assumptionassump:doeblin and that $\Phi$ is strictly positive. Then

Theorems & Definitions (28)

  • Lemma 2.2
  • proof
  • Proposition 2.3: Rank-one decomposition
  • Remark 2.4: Boundedness of operators
  • Lemma 3.1: Sherman--Morrison formula SM1950B1951
  • proof
  • Corollary 3.2: Pole and residue of the resolvent of a rank-one operator
  • Theorem 3.3: Birman--Schwinger resolvent factorization
  • proof : Proof.
  • Corollary 3.4: Residue at a simple zero of $D(\lambda)$
  • ...and 18 more