Determinants and traces of multidimensional discrete periodic operators with defects

TL;DR

This paper develops a trace and determinant theory for a class of multidimensional discrete periodic operators with defects, extending existing algebraic frameworks and providing explicit formulas for these invariants.

Contribution

It introduces a new algebraic structure for operators with defects and defines trace and determinant functions with desirable properties, generalizing previous theories.

Findings

Defined a trace $ au$ and a determinant $pi$ for the operator algebra

Proved $ au$ and $pi$ are continuous and analytic functions

Established the algebra contains multiplication operators and trace class operators

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.