We prove estimates on the speed of convergence of the ‘peripheral eigenvalues’ (and principal eigenvectors) of a sequence Tn of positive operators on a Banach lattice E to the peripheral eigenvalues of its limit operator T on E which is positive, irreducible and such that the spectral radius r(T) of T is a Riesz point of the spectrum of T (that is, a pole of the resolvent of T with a residuum of finite rank) under some conditions on the kind of approximation of Tn to T. These results sharpen results of convergence obtained by the authors in previous papers.

Let Σ be a countably generated left amenable semigroup and ﹛Tσ|σ ∈ Σ﹜ be a representation of Σ as a semigroup of positive linear operators on a weakly sequentially complete Banach lattice E with a weak unit e. It is assumed Tσ are uniformly bounded. It is shown that a necessary and sufficient condition for the existence of a weak unit invariant under ﹛Tσ | σ ∈ Σ﹜ is that inf σ∈Σ H(Tσe) > 0 for all nonzero H in the positive dual cone of E.

A vector lattice F is said to be Dedekind α-complete, where α is a cardinal number, provided that each non-empty order bounded subset D of F satisfying card(D) ≤ α has a supremum. Several characterizations of this property are presented here.

In this paper we study some aspects of the behaviour of p-lattice summing operators. We prove first that an operator T from a Banach space E to a Banach lattice X is p-lattice summing if and only if its bitranspose is. Using this theorem we prove a characterization for 1 -lattice summing operators defined on a C(K) space by means of the representing measure, which shows that in this case 1 -lattice and ∞-lattice summing operators coincide. We present also some results for the case 1 ≤ p < ∞ on C(K,E).

Let E be a Banach Lattice. We will consider E to be weakly sequentially complete and to have a weak unit u. Thus we may represent E as a lattice of real valued functions defined on a measure space (χ, , μ). There is a set R ⊂ χ such that R supports a maximal invariant function Φ for a postive contraction T on E [5]. Let N = χ — R be the complement of R. Akcoglu and Sucheston showed that where E+ is the positive cone of E. If in addition a monotone condition (UMB) is satisfied, then the same authors showed [4] that converges in norm.

We consider the Banach algebra consisting of linear operators T which are defined on the simple functions and have bounded extensions Tp on LP for all values of p ∊ [1, ∞]. We show that the 'integral' operators in this algebra form a right ideal, and that each Tp associated to an integral T is regular. When the underlying measure is finite or special discrete we show further that every Tp is regular for every T in the algebra. Algebraic techniques together with interpolation results are then used to get relationships between the spectrum and the order spectrum of the associated Tp's.

In the spirit of our previous paper (Math. Z. 156, 265 - 277 (1977)) we present a functional analytic proof of the following result of M. A. Akcoglu: Every positive contraction on a reflexive Lp-space has a lattice dilation.