On the Point Spectrum of Positive Operators

1. Recently, G.-C. Rota proved the following result: Let (S, 2, p) be a measure space of finite measure, P a positive linear operator on Lx(S, 2, u) with Li-norm and L„-norm at most one. If a, | a\ = 1, is an eigenvalue of P such that af=Pf (JELx), then a2 is an eigenvalue such that a2|/|g"=P(|/|g"), where/=|/|g. It can be added that an|/|gn = P(|/|gn) for every integer n; thus Rota proved for a fairly large class of operators, without compactness assumptions, a result known (and due to Frobenius) for positive finite square matrices, and known for certain types of positive operators under conditions guaranteeing that the spectrum intersects the circumference of the spectral circle but in a finite set (see Karlin [l, pp. 933-935] for an excellent survey and some more general examples). Simple but typical examples of operators showing the spectral behavior exhibited in Rota's theorem are the permutation matrices on lp ilépé <*). The purpose of this paper is to extend Rota's result to a larger class of spaces and operators. Apart from the particular type of underlying space, the stringent condition in Rota's theorem (supposing that ju(S) = 1) isr(7,)=||T'||i-=||r||0<>, r(T) denoting the spectral radius of T in Lx which is implicitly assumed to be one in [2]. From this, we can drop the total finiteness of ¿u, the assumption || 2~"||x = 11 T\\a and the requirement that Lx be invariant under T (T need indeed not be defined on all of L„ when ju(5) is infinite). More generally (Theorem 1), the result is true for positive operators on any complex function space E of type LPiS, 2, ju) or C(X) (A compact Hausdorff), whenever T'i¡/^\¡/ f°r some strictly positive linear form ipEE'.1 This class includes all quasi-interior positive operators on C(A), for which other spectral properties were obtained in [3]. More particularly, for positive matrix operators on lp satisfying the assumption above with respect to some strictly positive linear form, the presence (assuming r(T) = l) of a single unimodular eigenvalue which is not a root of unity, implies that the entire unit circle is in the point spectrum of T (Theorem 2). The assumption that T'xp^ij/ for some strictly positive linear form, in particular satisfied through ||r||i = l in Rota's theorem, is by no means necessary for the conclusion; it is made to ensure that