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Spectral properties of non-selfadjoint difference operators. (English) Zbl 0992.39018
The authors consider the operator $L$ generated in $\ell^2({\Bbb Z})$ by the difference expression $(\ell y)_n=a_{n-1}y_{n-1}+b_ny_n+a_ny_{n+1}$, $n\in{\Bbb Z}$, where $\{a_n\}_{n\in{\Bbb Z}}$ and $\{b_n\}_{n\in{\Bbb Z}}$ 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 $\ell({\Bbb Z,\Bbb C}^2)$ by the system of the difference expression $$ \pmatrix \Delta y_n^{(2)}+p_ny_n^{(1)}\cr -\Delta y_{n-1}^{(1)}+q_ny_n^{(2)} \endpmatrix, $$ $n\in{\Bbb Z}$, where $\{p_n\}_{n\in{\Bbb Z}}$ and $\{q_n\}_{n\in{\Bbb Z}}$ are complex sequences.

39A70Difference operators
39A12Discrete version of topics in analysis
34L05General spectral theory for OD operators
Full Text: DOI
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