论四元基本数值范围的凸性

Pub Date : 2024-05-15 DOI:10.1017/s0013091524000336

LuÍs Carvalho, Cristina Diogo, Sérgio Mendes, Helena Soares

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引用次数: 0

摘要

一般来说，四元数环境中的数值范围是四元数的一个非凸子集。本质数值范围是数值范围的细化，只保留在某种意义上具有无限倍性的元素。我们证明了四元数希尔伯特空间上有界线性算子的基本数值范围是凸的。我们还提供了兰卡斯特定理的四元数类似定理，它涉及数值范围的闭合及其基本数值范围。

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On the convexity of the quaternionic essential numerical range

The numerical range in the quaternionic setting is, in general, a non-convex subset of the quaternions. The essential numerical range is a refinement of the numerical range that only keeps the elements that have, in a certain sense, infinite multiplicity. We prove that the essential numerical range of a bounded linear operator on a quaternionic Hilbert space is convex. A quaternionic analogue of Lancaster theorem, relating the closure of the numerical range and its essential numerical range, is also provided.

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