On the diagonal of Riesz operators on Banach lattices

TL;DR

This paper generalizes the concept of the diagonal for Riesz operators on Banach lattices, establishing structural properties and identifying classes of operators where the diagonal coincides with the atomic diagonal.

Contribution

It introduces an abstract diagonal for regular operators on order complete vector lattices and shows that certain classes form bands, extending spectral theory for Riesz operators.

Findings

The class of operators with diagonal equal to the atomic diagonal forms a band.

This class includes all positive Riesz operators on Banach lattices.

The spectrum of ideal-triangularizable Riesz operators lies on their diagonal.

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 $\mathscr D$ of regular operators for which the diagonal coincides with the atomic diagonal is always a band in $\mathcal L_r(E)$, which contains the band of abstract integral operators. If $E$ is also a Banach lattice, then $\mathscr D$ contains positive Riesz operators.