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 ∈ C and y ∈ X are given, and (ii) the eigenvalue problem T ϕ = λϕ, where λ ∈ C and 0 = ϕ ∈ X , one approximates the operator T by a sequence (T n ) of bounded linear operators on X . We consider pointwise convergence, norm convergence, and nu convergence of (T n ) to T . We give several examples to illustrate possible scenarios. In most classical methods of approximation, each T n is of finite rank. We give a canonical procedure for reducing problems involving finite rank operators to problems involving matrix computations.