The classical Perron–Frobenius theory asserts that, for two matrices \documentclass[12pt]{minimal}
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\begin{document}$$A$$\end{document} and \documentclass[12pt]{minimal}
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\begin{document}$$B$$\end{document}, if \documentclass[12pt]{minimal}
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\begin{document}$$0\le B \le A$$\end{document} and \documentclass[12pt]{minimal}
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\begin{document}$$r(A)=r(B)$$\end{document} with \documentclass[12pt]{minimal}
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\begin{document}$$A$$\end{document} being irreducible, then \documentclass[12pt]{minimal}
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\begin{document}$$A=B$$\end{document}. It has been extended to infinite-dimensional Banach lattices under certain additional conditions, including that \documentclass[12pt]{minimal}
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\begin{document}$$r(A)$$\end{document} is a pole of the resolvent of \documentclass[12pt]{minimal}
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\begin{document}$$A$$\end{document}. In this paper, we prove that the same result holds if \documentclass[12pt]{minimal}
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\begin{document}$$B$$\end{document} is irreducible and \documentclass[12pt]{minimal}
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\begin{document}$$r(B)$$\end{document} is a pole of the resolvent for \documentclass[12pt]{minimal}
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\begin{document}$$B$$\end{document}. We also prove some other interesting extensions of the theorem for infinite-dimensional Banach lattices.
