On principal T-bands in a Banach lattice

It is shown that for every positive order continuous Riesz operator T, defined on an order complete complex Banach lattice E which is separated by its Kothe dual; there exists a Frobenius decomposition of E into a countable number of disjoint principal T-bands and a band on which T is quasi-nilpotent. A basis for the generalized eigenspace of T pertaining to its maximal eigenvalue is constructed and the positivity properties of its elements are studied. The distinguished eigenvalues of T are characterized and it is also shown that the theory of T-bands is symmetric with respect to the duality which exists between E and its Kothe dual. This generalizes aspects of work done by H.D. Victory and R.-J. Jang-Lewis.