Compact groups of positive operators on Banach lattices

Abstract

Abstract. In this paper we study groups of positive operators on Banach lattices. If a certain factorization property holds for the elements of such a group, the group has a homomorphic image in the isometric positive operators which has the same invariant ideals as the original group. If the group is compact in the strong operator topology, it equals a group of isometric positive operators conjugated by a single central lattice automorphism, provided an additional technical assumption is satisfied, for which we have only examples. We obtain a characterization of positive representations of a group with compact image in the strong operator topology, and use this for normalized symmetric Banach sequence spaces to prove an ordered version of the decomposition theorem for unitary representations of compact groups. Applications concerning spaces of continuous functions are also considered.

Convergence of positive operator semigroups

“…Some spectral theoretic results about positive group representations can be found in [53,24] and some structural results about groups of positive operators in [48,Sec 3]. A systematic investigation of positive group representations was just initiated in the recent series of papers [11,12,10]. While all of these references focus on strongly continuous representations of certain topological groups, we do not require any topological assumptions in the following.…”