Spectral properties of compact lattice homomorphisms

Wickstead, Anthony W.

doi:10.1090/S0002-9939-1982-0640228-8

Spectral properties of compact lattice homomorphisms
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Proc. Amer. Math. Soc. 84 (1982), 347-353 Request permission

Abstract:

Given two nonzero eigenvalues of a lattice homomorphism on a relatively uniformly complete vector lattice, of different moduli and with at least one isolated in the set of all eigenvalues, we show that corresponding eigenvectors must be disjoint. The analogous result for the approximate point spectrum of a lattice homomorphism on a Banach lattice is deduced. We give an infinite spectral decomposition for a lattice homomorphism, on a Banach lattice with order continuous norm, which is compact and has an adjoint which is also a lattice homomorphism. From this we deduce that if it has nonnegative spectrum, then it is the direct sum of a nilpotent lattice homomorphism and one that is central.

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