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

Abstract

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 finite rank and a partition of unity property. Then all the points in the spectrum are eigenvalues of $T$ and ${\sigma }_p(T)\subset B(0,1)\cup \{1\}$. The main result is that $1$ is the only eigenvalue on the unit circle, the peripheral spectrum of $T$.