On the extension of Fredholm determinants to the mixed multidimensional integral operators with regulated kernels

We extend the classical trace (and determinant) known for the integral operators $$({\mathcal I}+)\int_{[0,1)^N}{\bf A}({\bf k},{\bf x}){\bf u}({\bf x})d{\bf x}$$ with matrix-valued kernels ${\bf A}$ to the operators of the form $$\sum_{\alpha}\int_{[0,1)^{|\alpha|}}{\bf A}({\bf k},{\bf x}_{\alpha}){\bf u}({\bf k}_{\overline{\alpha}},{\bf x}_{\alpha})d{\bf x}_{\alpha},$$ where $\alpha$ are arbitrary subsets of the set $\{1,...,N\}$. Such operators form a Banach algebra containing simultaneously all integral operators of the dimensions $\leqslant N$. In this sense, it is a largest algebra where explicit traces and determinants are constructed. Such operators arise naturally in the mechanics and physics of waves propagating through periodic structures with various defects. We give an explicit representation of the inverse operators (resolvent) and describe the spectrum by using zeroes of the determinants. Due to the structure of the operators, we have $2^N$ different determinants, each of them describes the spectral component of the corresponding dimension.