AI 中文总结
本文提出适配线性关系多值特性的巴拿赫空间数值域新定义，推导相关谱包含结果与预解估计，将其应用于算子束得到更精确的谱信息，还可用于半群的严格指数衰减分析。
AI 中文摘要
本文致力于研究巴拿赫空间中线性关系的数值域。我们提出了一种适配线性关系多值特性的新定义，并分析其主要性质。我们建立了谱包含结果，表明谱包含于线性关系数值域的闭包与其巴拿赫伴随的数值域的并集，以及与到数值域距离相关的预解估计。作为应用，我们通过将算子束与合适的线性关系关联，推导了算子束的谱包络，并引入了巴拿赫空间中算子束对应的数值域。我们表明，该方法可提供比算子束经典数值域更精确的信息。此外，我们利用数值横坐标证明，巴拿赫空间数值域可对无法从相应希尔伯特空间数值域得到的半群产生严格指数衰减。
英文摘要
This paper is devoted to the study of the numerical range of linear relations in Banach spaces. We present a new definition adapted to the multivalued nature of linear relations and analyze its main properties. We establish spectral inclusion results showing that the spectrum is contained in the union of the closure of the numerical range of a linear relation and the numerical range of its Banach adjoint, together with resolvent estimates related to the distance to the numerical range. As an application, we derive spectral enclosures for operator pencils by associating them with suitable linear relations and introduce corresponding numerical ranges for operator pencils in Banach spaces. We show that this approach may provide sharper information than classical numerical ranges of operator pencils. Furthermore, we use the numerical abscissas to show that the Banach space numerical range can yield strict exponential decay for semigroups that cannot be obtained from the corresponding Hilbert space numerical range.
