What This Book Is About: Approximants

Abstract

The key concept of this book is that of an approximant (the characteristically snappy term is due to Halmos [21]). Let $\mathbb{L}$, say, be a space of mathematical objects (complex numbers or square matrices, say); let $\mathbb{N}$ be a subset of $\mathbb{L}$ each of whose elements have some “nice” property p (of being real or being self-adjoint, say); and let A be some given, not nice element of $\mathbb{L}$; then a p-approximant of A is a nice mathematical object that is nearest, with respect to some norm, to A. In the first example just mentioned, a given complex number z has its real part $\mathbb{R}z(={ z+\bar{z} \over 2} )$ as its (unique) real approximant. In the second example, a given square matrix A has (by Theorem 3.2.1) its real part $\mathbb{R}A(={ A+A^{{\ast}} \over 2} )$ as its unique self-adjoint approximant.