The Upper Bounds for Powers of Linear Operators and Some Applications to the Stability Analysis of Difference Problems

The paper is devoted to deriving the upper bounds for powers of linear operatorsin Banach norms. The main assumption is the Kreiss resolvent condition in its gneralized form. The present paper makes an emphasis on the case. when an input oprator is a function of another operator. Together with the standard bounds, those of powers of linear operators in composition with smoothing operators are established. The applications to the stability theory of difference problems are considered.