Positive Solutions of Positive Linear Equations

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 A is a positive real. It is shown that (XI—TY1y is positive if, and only if, y is annihilated by the absolute value of any generalized eigenvector of T* associated with a strictly positive eigenvalue not less than /. A strictly positive eigenvalue is a positive eigenvalue having an associated positive eigenvector. For the case of B=L" this yields the result that (A/— T)~ly±i0 if, and only if, y is almost everywhere zero on a certain set which depends on X but is otherwise fixed. In some fields of applied mathematics (e.g., radiative transfer, neutron transport) there occur conditional equations of the form