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

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.

💡 Research Summary

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The paper develops a determinant‑free, resolvent‑based framework for characterizing the dominant spectral pole of positive operators on Banach lattices under a rank‑one Doeblin‑type minorization. Classical results such as the Perron–Frobenius theorem or the Krein–Rutman theorem typically require strong positivity, compactness, or trace‑class assumptions to guarantee a positive, simple leading eigenvalue and an associated rank‑one spectral projection. By contrast, the authors assume only that the operator T admits a lower bound of the form
    K(x,y) ≥ α u₀(x) g(y) for all x,y,
with α>0, u₀∈E₊, and a strictly positive functional Φ