I am exploring the concept of Parseval's identity in the context of matrix-valued functions. While Parseval's identity is well-known for scalar-valued functions: $$ \int_{\mathbb{T}^3} f(x) \cdot \overline{g(x)} \, dx = \sum_{k\in\mathbb{Z}^3} \hat{f}_k\cdot \overline{\hat{g}_k}.$$ Even if $f$ and $g$ are vectors, one can use the dot product. I am interested in understanding how it can be adapted or extended to matrix-valued functions.
Specifically, I am working with periodic matrix-valued functions defined over the 3D torus $\mathbb{T}^3$. I would like to establish a formulation of Parseval's identity that applies to such functions, considering their periodic nature.
Can anyone provide insights, references, or a mathematical framework for Parseval's identity when dealing with periodic matrix-valued functions?