Some spectral properties of positive linear operators

It is well known (Perron [12], Frobenius [6, 7]) that if A is an n x n matrix over the real field with elements ^ 0, the spectral radius 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. 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]. While the key to many results generalizing the Perron-Frobenius theorem is compactness in one form or another, a good many spectral properties of positive linear operators are independent of it. Such properties were established by Bonsall (e.c, cf. Prop. 1 below), the author 117], and recently Putnam [13] who considers, however, only the rather special case of a bounded matrix with non-negative elements in l2. The present paper establishes new and more general results on the (spectral) character of the spectral radius r of a positive operator T, valid in arbitrary ordered Banach spaces. Section 2 collects some theorems for which no hypothesis or r is made; leaning heavily on topological properties of the positive cone K, they apply to any positive operator. Throughout § 3, r is assumed to be a pole of the resolvent of T. The stress is here on the notion of quasi-interior map; together with the assumption on r, this concept yields strong results earlier obtained by Krein and Rutman [11] for strongly positive operators which are compact and defined on a space whose positive cone K has interior points. This is interesting since in many concrete examples of partially ordered (B)-spaces, K has empty interior [16, p. 130]. The paper concludes with two problems.