Operator Approximation

Abstract

We present an introduction to operator approximation theory. Let T be a bounded linear operator on a Banach space X over ${\mathbb C}$. In order to find approximate solutions of (i) the operator equation $z\,x-Tx=y$, where $z\in {\mathbb C}$ and $y\in X$ are given, and (ii) the eigenvalue problem $T\varphi =\lambda \varphi$, where $\lambda \in {\mathbb C}$ and $0\ne \varphi \in X$, 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.