Optimal bounds for bilinear forms associated with linear equations

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.