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Spectral properties of non-selfadjoint difference operators. (English) Zbl 0992.39018

The authors consider the operator $L$ generated in ${\ell }^{2}\left(ℤ\right)$ by the difference expression ${\left(\ell y\right)}_{n}={a}_{n-1}{y}_{n-1}+{b}_{n}{y}_{n}+{a}_{n}{y}_{n+1}$, $n\in ℤ$, where ${\left\{{a}_{n}\right\}}_{n\in ℤ}$ and ${\left\{{b}_{n}\right\}}_{n\in ℤ}$ are complex sequences. The spectrum, the spectral singularities, and the properties of the principal vectors corresponding to the spectral singularities of $L$ are investigated. The authors also study similar problems for the discrete Dirac operator generated in $\ell \left({ℤ,ℂ}^{2}\right)$ by the system of the difference expression

$\left(\begin{array}{c}{\Delta }{y}_{n}^{\left(2\right)}+{p}_{n}{y}_{n}^{\left(1\right)}\\ -{\Delta }{y}_{n-1}^{\left(1\right)}+{q}_{n}{y}_{n}^{\left(2\right)}\end{array}\right),$

$n\in ℤ$, where ${\left\{{p}_{n}\right\}}_{n\in ℤ}$ and ${\left\{{q}_{n}\right\}}_{n\in ℤ}$ are complex sequences.

##### MSC:
 39A70 Difference operators 39A12 Discrete version of topics in analysis 34L05 General spectral theory for OD operators