Discreteness and estimates of spectrum of a first order difference operator

Abstract

We investigated a minimal closed in the space l₂ 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.