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We characterize the spectrum of positive linear operators $T:X \to Y$, where$X$ and $Y$ are complex Banach function spaces with unit $1$, having finiterank and a partition of unity property. Then all the points in the spectrum areeigenvalues of $T$ and $\sigma_p(T) \subset B(0,1) \cup \{1\}$. The main resultis that $1$ is the only eigenvalue on the unit circle, the peripheral spectrumof $T$.

On the spectrum of positive finite-rank operators with a partition of unity property