Math. Z. 209, 547 558 (1992) Mathematische Zeitschrift ~) Springer-Verlag 1992 Approximations of positive operators and continuity of the spectral radius II F. Ardndiga 1 and V. Caselles 2 1 Facultat de Matem~tiques, C/Doctor Moliner, 50, E-46100 Burjassat (Valencia), Spain 2 Facult6 de Mathematiques, Universit6 de Franche-Comt6 U.A. CNRS 741, Route de Gray s/n, F-25030 Besan~on Cedex, France Received October 24, 1990; in final form July 5, 1991 I Introduction In our paper [1], we set up a framework to study the convergence of some approximate eigenvalues and their eigenvectors of positive approximations of positive operators to the peripheral eigenelements of the (positive) limit problems. Let us recall our main result in [1]. Let 0<T., T be bounded, linear operators on a Banach lattice E such that T, order converges to T (or converges uniformly on the order intervals of E) and [I(T,--T)+II--*0 as n--,0. Let us suppose that T is an irreducible operator on E such that r(T) is a Riesz point of a(T). Suppose also that T= T1 + T2 with T2>0 being an AM-compact operator. Then, r(T,)~r(T), some approximate eigenvalues (spectrum) of T, converges to the peripheral spectrum of T and a similar statement is true for the eigenvectors [1, Theorems 3.1, 3.3 and 3.6]. We have deliberately omitted one assumption in the above statement: E is a dual Banach lattice with order continuous norm (notice that this implies that E is weakly sequentially complete). Thus, the case when E is an Ll-space was not covered by our results in [1]. It is our purpose here to fill this gap. We prove a result similar to stated aboved if E is a weakly sequentially complete Banach lattice (hence, including Ll-spaces). On the other hand, the decomposition required on T such that T=TI+T2 has to be restricted a little bit. T/>0 will be an abstract kernel operator (hence AM-compact [14, Theorem 123.9]). This is of no harm in concrete situations. Let us finally mention that our general strategy is based on an interpolation argument. We construct a Banach lattice F contained in F such that F is a dual Banach lattice with order continuous norm, T(F) c F and the peripheral spectrum of the restriction of T to F coincides with the peripheral spectrum of T. Thus, the restriction of T to F satisfies the assumptions of our previous results. Then, after using them, we conclude by going back again to E and getting the correspondent consequences for T., T as operators on E. Let us start by recalling some notation and terminology.