Positive solutions of positive linear equations

Abstract:

Let $B$ be a real vector lattice and a Banach space under a semimonotonic norm. Suppose $T$ is a linear operator on $B$ which is positive and eventually compact, $y$ is a positive vector, and $\lambda$ is a positive real. It is shown that ${(\lambda I - T)^{ - 1}}y$ is positive if, and only if, $y$ is annihilated by the absolute value of any generalized eigenvector of ${T^\ast }$ associated with a strictly positive eigenvalue not less than $\lambda$. A strictly positive eigenvalue is a positive eigenvalue having an associated positive eigenvector. For the case of $B = {L^p}$ this yields the result that ${(\lambda I - T)^{ - 1}}y \geqq 0$ if, and only if, $y$ is almost everywhere zero on a certain set which depends on $\lambda$ but is otherwise fixed.