The present work sets forth the analytical tools which make it possible to construct and compute invariant kernels on the space of complex covariance matrices. The main result is the $\mathrm{L}^1$--\,Godement theorem, which is used to obtain a general analytical expression that yields any invariant kernel which is, in a certain natural sense, also integrable. Using this expression, one can design and explore new families of invariant kernels, all while incurring a rather moderate computational effort. The expression comes in the form of a determinant (it is a determinantal expression), and is derived from the notion of spherical transform, which arises when the space of complex covariance matrices is considered as a Riemannian symmetric space.

翻译：本文提出了在复协方差矩阵空间上构造和计算不变核的分析工具。主要结果是$\mathrm{L}^1$--Godement定理，该定理用于获得一个通用解析表达式，该表达式能够生成任何在自然意义下可积的不变核。利用此表达式，可以设计和探索新的不变核族，同时仅需适中的计算成本。该表达式以行列式形式呈现（即为行列式表达式），其推导基于球面变换的概念，而球面变换源于将复协方差矩阵空间视为黎曼对称空间。