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Non-self-adjoint difference operators and their spectrum. (English) Zbl 1145.47303
Summary: Initially, this paper is a discrete analogue of the work of B. M. Brown, D. K. R. McCormack, W. D. Evans and M. Plum [Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 455, No. 1984, 1235–1257 (1999; Zbl 0944.34018)] on second-order differential equations with complex coefficients. That is, we investigate the general non-self-adjoint second-order difference expression ${M}_{{x}_{n}}=-{\Delta }\left({p}_{n-1}{\Delta }{x}_{n-1}\right)+{q}_{n}{x}_{n}$, $n\in {N}_{0}$ where the coefficients ${p}_{n}$ and ${q}_{n}$ are complex and ${\Delta }$ is the forward difference operator, i.e. ${\Delta }{x}_{n}={x}_{n+1}-{x}_{n}$. Properties of the so-called Hellinger-Nevanlinna m-function for the recurrence relation ${M}_{{x}_{n}}=\lambda {w}_{n}{x}_{n}$, where the ${w}_{n}$ are real and positive, are examined, and relationships between the properties of the $m$-function and the spectrum of the associated operator are explored. However, an essential difference between the continuous and the discrete case arises in the way in which we define the operator natural to the problem. Nevertheless, analogous results regarding the spectrum of this operator are obtained.

##### MSC:
 47B39 Difference operators (operator theory) 39A70 Difference operators 47A10 Spectrum and resolvent of linear operators