An Inequality for Some Nonnormal Operators

Abstract

An inequality of use in testing convergence of eigenvector calculations is improved. If ${e_\lambda }$ is a unit eigenvector corresponding to an eigenvalue $\lambda$ of a dominant operator $A$ on a Hilbert space $H$, then \[ |(g,{e_\lambda }){|^2} \leq \frac {{||g|{|^2}||Ag|{|^2} - |(g,Ag){|^2}}}{{||(A - \lambda I)g|{|^2}}}\] for all $g$ in $H$ for which $Ag \ne \lambda g$. The equality holds if and only if the component of $g$ orthogonal to ${e_\lambda }$ is also an eigenvector of $A$. This result is an improvement of Bernstein’s result for selfadjoint operators.