A bound on powers of linear operators, with relevance to numerical stability
The authors consider a bounded linear Banach space operator \(T\) with spectrum in the closed unit disk. It is assumed that there are two integers \(k,m\) such that \((I-zT)^{-k-1}=H_1(z)+\cdots+H_m(z)\) in the open unit disk, where \(H_j\) are operator-valued holomorphic functions with certain growth restrictions. The main result is an upper bound for the norms of the consecutive powers: \(\|T^n\|\leq C_m/{n+k \choose k}\), where the constant \(C_m\) can be specified in terms of \(H_j\). The authors then give an application of this result to the operator \(T\) of the Crank-Nicolson finite difference method for the numerical solution of the heat equation. The application yields a new stability estimate for \(\|T^n\|\).
scientific article; zbMATH DE number 1878830