Operator approximation

Abstract

We present an introduction to operator approximation theory. Let T be a bounded linear operator on a Banach space X over ℂ. In order to find approximate solutions of (i) the operator equation z x − T x = y, where z ϵ ℂ and y ϵ X are given, and (ii) the eigenvalue problem Tϕ = λϕ, where λ ϵ ℂ and 0 ≠ ϕ ϵ X, one approximates the operator T by a sequence (Tn) of bounded linear operators on X. We consider pointwise convergence, norm convergence, and nu convergence of (Tn) to T. We give several examples to illustrate possible scenarios. In most classical methods of approximation, each Tn is of finite rank. We give a canonical procedure for reducing problems involving finite rank operators to problems involving matrix computations.