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Short communications

Discreteness and estimates of spectrum of a first order difference operator

K. N. Ospanov

Department of Mechanics and Mathematics, L.N. Gumilyov Eurasian National University, 13 Munaitpasov St, 010008 Astana, Kazakhstan

Abstract: We investigated a minimal closed in the space $l_2$ first order nonsymmetric difference operator $L$. The matrix of zero order coefficients of $L$ may be an unbounded operator. The study of $L$ is motivated by applications to stochastic processes and stochastic differential equations. We obtained compactness conditions and exact with respect to the order two-sided estimates for $s$-numbers of the resolvent of $L$. Note that these estimates for $s$-numbers do not depend on the oscillations of the coefficients of $L$, in contrast to the case of a differential operator.

Keywords and phrases: difference operator, coercive estimate, compactness of the resolvent, singular numbers.

MSC: 39A70, 47B39

Received: 12.06.2017

Language: English