2015/10/20 by Anton A. Kutsenko, Kutsenko, Anton A.
Mathematics · Physics and Astronomy · #FOS: Physical sciences #Mathematical Physics (math-ph) #math-ph #math.MP
paper · pdf · doi:10.48550/arxiv.1510.05906
arxiv created 2015/10/24 · arxiv updated 2015/10/27
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 A0\bf u+\bf A1∫01dk1\bf B1\bf u+...+\bf AN∫01dk1...∫01dkN\bf BN\bf u, \bf u∈ L2([0,1]N,ℂM), where (\bf A,\bf B)(k1,...,kN) are continuous matrix-valued functions of appropriate sizes. All such operators form a non-closed algebra \mathscr HN,M. In this article we show that there exist a trace \pmbτ and a determinant \pmbπ defined for operators from \mathscr HN,M with the properties \pmbτ(α\mathcal A+β\mathcal B)=α\pmbτ(\mathcal A)+β\pmbτ(\mathcal B), \pmbτ(\mathcal A\mathcal B)=\pmbτ(\mathcal B\mathcal A), \pmbπ(\mathcal A\mathcal B)=\pmbπ(\mathcal A)\pmbπ(\mathcal B), \pmbπ(e\mathcal A)=e^\pmbτ(\mathcal A). The mappings \pmbπ, \pmbτ are vector-valued functions. While \pmbπ has a complex structure, \pmbτ is simple \pmbτ(\mathcal A)=(\rm Tr\bf A0,∫01dk1\rm Tr\bf B1\bf A1,...,∫01dk1...∫01dkN\rm Tr\bf BN\bf AN). There exists the norm under which the closure \mathscr HN,M is a Banach algebra, and \pmbπ, \pmbτ 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.