The spectrum of operator extensions to free Banach Lattices

Jochen Glück; Phillip Krokor

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arXiv · 2608.11437

The spectrum of operator extensions to free Banach Lattices

Jochen Glück·Phillip Krokor

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Abstract

Every bounded linear operator $T$ on a complex Banach space $E$ is known to extend to a lattice homomorphism $\overline{T}$ that acts on the so-called complex free Banach lattice over $E$. We show the following three results about the spectrum $\sigma(\overline{T})$ of $\overline{T}$: (i) $\sigma(\overline{T})$ always contains the spectrum $\sigma(T)$; this answers a recent question of de Hevia and Tradacete. (ii) It can happen that $\sigma(T)$ is a singleton while $\sigma(\overline{T})$ is the entire unit circle; this shows that $\sigma(\overline{T})$ is not the closure of the cyclic hull of $\sigma(T)$ in general. (iii) Finally, we give a full characterization of $\sigma(\overline{T})$ in terms of the spectral properties of $T$. Some of our arguments also give new results about the spectral properties of general lattice homomorphisms. As a main tool we make extensive use of the Banach lattice functional calculus for continuous positively homogeneous functions.

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Jochen Glück, Phillip Krokor. 2026-08-11. The spectrum of operator extensions to free Banach Lattices. https://arxiv.org/abs/2608.11437

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