Optimal bounds for bilinear forms associated with linear equations
J. Mika, D. C. Pack, Richard J. Cole, Gary F. Roach · Mathematical Methods in the Applied Sciences · 1985
Abstract The construction of upper and lower bounds to the bilinear quantity g0, f, where f is the solution of an operator equation Tf = f0, requires either an approximation for f or one for T−1. In this paper the question of “best” approximation of T−1 by an operator of the form B = βI, where β is a real constant, is investigated for linear operators that are either self‐adjoint or can be related by suitable manipulations to others that are. Particular attention is paid to a special operator, previously studied by Robinson, of importance in predicting the dynamic polarisabilities of quantum‐mechanical systems.