Invariant subspaces for positive operators on Banach spaces with unconditional basis

Gallardo-Gutiérrez, Eva; González-Doña, Javier; Tradacete, Pedro

doi:10.1090/proc/16026

Invariant subspaces for positive operators on Banach spaces with unconditional basis
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by Eva A. Gallardo-Gutiérrez, Javier González-Doña and Pedro Tradacete PDF

Proc. Amer. Math. Soc. 150 (2022), 5231-5242 Request permission

Abstract:

We prove that every lattice homomorphism acting on a Banach space $\mathcal {X}$ with the lattice structure given by an unconditional basis has a non-trivial closed invariant subspace. In fact, it has a non-trivial closed invariant ideal, which is no longer true for every positive operator on such a space. Motivated by these examples, we characterize tridiagonal positive operators without non-trivial closed invariant ideals on $\mathcal {X}$ extending to this context a result of Grivaux on the existence of non-trivial closed invariant subspaces for tridiagonal operators.

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