An extremal property of positive operators and its spectral implications

Abstract

The Perron-Frobenius theory of a positive operator T (defined on an ordered space E) is developed from an extremal characterization of its largest eigenvalue. This characterization has manifest intrinsic interest. Additionally, it is used to give a particularly revealing derivation of the basic results concerning the existence, multiplicity, and distribution of the eigenvalues of T of maximum modulus. A significant feature of this derivation is that the customary assumptions that the space E be complete and/or that its positive cone have a nonvoid interior are often unnecessary or can be replaced by weaker hypotheses more amenable to practical applications (see 1, 3). The extremal characterization proof of the distribution properties of the eigenvalues of maximum modulus is new in the infinite dimensional case. Also, several new results on the extent of applicability of the extremal characterization are given