  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.
