The spectrum of operator extensions to free Banach Lattices
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.