Perturbation analysis has emerged as a significant concern across multiple disciplines, with notable advancements being achieved, particularly in the realm of matrices. This study centers on specific aspects pertaining to tensor T-eigenvalues within the context of the tensor-tensor multiplication. Initially, an analytical perturbation analysis is introduced to explore the sensitivity of T-eigenvalues. In the case of third-order tensors featuring square frontal slices, we extend the classical Gershgorin disc theorem and show that all T-eigenvalues are located inside a union of Gershgorin discs. Additionally, we extend the Bauer-Fike theorem to encompass F-diagonalizable tensors and present two modified versions applicable to more general scenarios. The tensor case of the Kahan theorem, which accounts for general perturbations on Hermite tensors, is also investigated. Furthermore, we propose the concept of pseudospectra for third-order tensors based on tensor-tensor multiplication. We develop four definitions that are equivalent under the spectral norm to characterize tensor $\varepsilon$-pseudospectra. Additionally, we present several pseudospectral properties. To provide visualizations, several numerical examples are also provided to illustrate the $\varepsilon$-pseudospectra of specific tensors at different levels.

翻译：扰动分析已成为多学科领域的重要关注点，尤其在矩阵领域取得了显著进展。本研究聚焦于张量-张量乘法框架中张量T-特征值的若干特定方面。首先，我们引入解析扰动分析方法探究T-特征值的敏感性。针对具有方形正面切片的三阶张量，我们将经典Gershgorin圆盘定理进行推广，证明所有T-特征值均位于若干Gershgorin圆盘的并集内。此外，我们将Bauer-Fike定理推广至可F对角化的张量，并给出适用于更一般情形的两种修正形式。同时研究了包含Hermite张量一般扰动的Kahan定理的张量情形。进一步地，我们基于张量-张量乘法提出三阶张量伪谱的概念，建立谱范数等价的四种定义以刻画张量ε-伪谱，并给出若干伪谱性质。为便于可视化，我们提供若干数值算例展示不同级别下特定张量的ε-伪谱。