Some spectral properties of positive linear operators

Abstract

It is well known (Perron [12], Frobenius [6, 7]) that if A is an n × n matrix over the real field with elements ≧ 0, the spectral radius1 of A, r(A), is a characteristic number, with at least one characteristic vector whose coordinates are ≧ 0. If A has positive elements throughout, then r is > 0, of algebraic and geometric multiplicity one, and exceeds all other elements of the spectrum in absolute value.2 Generalizations of this theorem to integral equations were obtained by Jentzsch [9] and E. Hopf [8]. In an operator-theoretic setting, the result did not appear until 1948 when Krein and Rutman published their most comprehensive work [11]. Further results were obtained by Bonsall [2]-[4] and, in the framework of a general locally convex space, by the author [15, 17] For compact positive operators in an order-complete Banach lattice, see Ando [1]. © 1960 by Pacific Journal of Mathematics.