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：扰动分析已成为多个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-张量乘法框架下张量T-特征值的特定方面。首先，引入解析扰动分析方法以探究T-特征值的敏感性。针对具有方阵正面切片的三阶张量，推广了经典Gershgorin圆盘定理，证明所有T-特征值均位于若干Gershgorin圆盘的并集内。此外，将Bauer-Fike定理扩展至适用于F-可对角化张量，并提出两个适用于更一般情形的改进版本。同时探讨了针对Hermite张量一般扰动的Kahan定理的张量情形。基于张量-张量乘法，提出三阶张量的伪谱概念，建立了在谱范数下等价的四种特征化张量ε-伪谱的定义，并给出若干伪谱性质。为直观展示，通过数值算例呈现不同扰动水平下特定张量的ε-伪谱分布。