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If $\\lambda_1^2>\\lambda_2$, then we show that $\\lambda_1$ is a simple eigenvalue of $f^*:H^{1,1}(X)\\rightarrow H^{1,1}(X)$, and moreover the unique eigenvalue of modulus $>\\sqrt{\\lambda_2}$. A variant of the result, where we consider the first spectral radius in the case the map $f$ may not be 1-stable, is also given. 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