On the diagonal of Riesz operators on Banach Lattices

Abstract

This paper extends the well-known Ringrose theory for compact operators to polynomially Riesz operators on Banach spaces. The particular case of an ideal-triangularizable Riesz operator on an order continuous Banach lattice yields that the spectrum of such  operator lies on its diagonal, which motivates the systematic study of an abstract diagonal of a regular operator on an order complete  vector lattice E. We prove that the class D of regular operators for which the diagonal coincides with the atomic diagonal is always a band  in Lr(E), which contains the band of abstract integral operators. If E is also a Banach lattice, then D contains positive Riesz and positive  AM-compact operators.