On the diagonal of Riesz operators on Banach lattices

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 motiva...

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Published in:Quaestiones mathematicae 2024-03, Vol.47 (sup1), p.137-151
Main Authors: Drnovšek, R., Kandić, M.
Format: Article
Language:English
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Banach lattices
Banach spaces
diagonal of an operator
Lattices (mathematics)
Linear operators
Operators (mathematics)
Riesz operators
Vector lattices
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description 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 of regular operators for which the diagonal coincides with the atomic diagonal is always a band in , which contains the band of abstract integral operators. If E is also a Banach lattice, then contains positive Riesz and positive AM-compact operators.
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subjects Banach lattices
Banach spaces
diagonal of an operator
Lattices (mathematics)
Linear operators
Operators (mathematics)
Riesz operators
Vector lattices
title On the diagonal of Riesz operators on Banach lattices
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