Bounds for the Rayleigh Quotient and the Spectrum of Self-Adjoint Operators

Abstract

The absolute change in the Rayleigh quotient (RQ) is bounded in this paper in terms of the norm of the residual and the change in the vector. If $x$ is an eigenvector of a self-adjoint bounded operator $A$ in a Hilbert space, then the RQ of the vector $x$, denoted by $\rho(x)$, is an exact eigenvalue of $A$. In this case, the absolute change of the RQ $|\rho(x)-\rho(y)|$ becomes the absolute error in an eigenvalue $\rho(x)$ of $A$ approximated by the RQ $\rho(y)$ on a given vector $y.$ There are three traditional kinds of bounds of the eigenvalue error: a priori bounds via the angle between vectors $x$ and $y$; a posteriori bounds via the norm of the residual $Ay-\rho(y)y$ of vector $y$; mixed type bounds using both the angle and the norm of the residual. We propose a unifying approach to prove known bounds of the spectrum, analyze their sharpness, and derive new sharper bounds. The proof approach is based on novel RQ vector perturbation identities.