AI 中文总结
该研究围绕复自由Banach格上算子延拓的谱展开，证明其包含原算子谱等三个结果，给出谱的完整刻画，还得到一般格同态谱性质的新结论。
AI 中文摘要
已知复Banach空间E上的每个有界线性算子T都可延拓为作用在E上的所谓复自由Banach格上的格同态T̄。我们证明了关于T̄的谱σ(T̄)的以下三个结果：(i) σ(T̄)始终包含谱σ(T)；这解答了de Hevia与Tradacete近期提出的一个问题。(ii) 可能出现σ(T)为单点集而σ(T̄)为整个单位圆周的情况；这表明σ(T̄)一般而言不是σ(T)的循环包的闭包。(iii) 最后，我们根据T的谱性质给出了σ(T̄)的完整刻画。我们的部分论证还得到了关于一般格同态谱性质的新结果。作为主要工具，我们大量使用了针对连续正齐次函数的Banach格函数演算。
英文摘要
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 $σ(\overline{T})$ of $\overline{T}$: (i) $σ(\overline{T})$ always contains the spectrum $σ(T)$; this answers a recent question of de Hevia and Tradacete. (ii) It can happen that $σ(T)$ is a singleton while $σ(\overline{T})$ is the entire unit circle; this shows that $σ(\overline{T})$ is not the closure of the cyclic hull of $σ(T)$ in general. (iii) Finally, we give a full characterization of $σ(\overline{T})$ 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.
Comments25 pages
