Determinants and traces of multidimensional discrete periodic operators with defects

Abstract: As it is shown in previous works, discrete periodic operators with defects are unitarily equivalent to the operators of the form $$ {\mathcal A}{\bf u}={\bf A}0{\bf u}+{\bf A}_1\int_0^1dk_1{\bf B}_1{\bf u}+...+{\bf A}_N\int_0^1dk_1...\int_0^1dk_N{\bf B}_N{\bf u},\ \ {\bf u}\in L^2([0,1]^N,\mathbb{C}^M), $$ where $({\bf A},{\bf B})(k_1,...,k_N)$ are continuous matrix-valued functions of appropriate sizes. All such operators form a non-closed algebra ${\mathscr H}{N,M}$. In this article we show that there exist a trace $\pmb{\tau}$ and a determinant $\pmb{\pi}$ defined for operators from ${\mathscr H}{N,M}$ with the properties $$ \pmb{\tau}(\alpha{\mathcal A}+\beta{\mathcal B})=\alpha\pmb{\tau}({\mathcal A})+\beta\pmb{\tau}({\mathcal B}),\ \ \pmb{\tau}({\mathcal A}{\mathcal B})=\pmb{\tau}({\mathcal B}{\mathcal A}),\ \ \pmb{\pi}({\mathcal A}{\mathcal B})=\pmb{\pi}({\mathcal A})\pmb{\pi}({\mathcal B}),\ \ \pmb{\pi}(e^{{\mathcal A}})=e^{\pmb{\tau}({\mathcal A})}. $$ The mappings $\pmb{\pi}$, $\pmb{\tau}$ are vector-valued functions. While $\pmb{\pi}$ has a complex structure, $\pmb{\tau}$ is simple $$ \pmb{\tau}({\mathcal A})=\left({\rm Tr}{\bf A}_0,\int_0^1dk_1{\rm Tr}{\bf B}_1{\bf A}_1,...,\int_0^1dk_1...\int_0^1dk_N{\rm Tr}{\bf B}_N{\bf A}_N\right). $$ There exists the norm under which the closure $\overline{{\mathscr H}}{N,M}$ is a Banach algebra, and $\pmb{\pi}$, $\pmb{\tau}$ are continuous (analytic) mappings. This algebra contains simultaneously all operators of multiplication by matrix-valued functions and all operators from the trace class. Thus, it generalizes the other algebras for which determinants and traces was previously defined.