# 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.

## Bibliography

- 1.J.G. Aiken, J.A. Erdos, J.A. Goldstein, Unitary approximation of positive operators. Ill. J. Math.
**24**, 61–72 (1980)MathSciNetzbMATHGoogle Scholar - 21.P.R. Halmos, Positive approximants of operators. Indiana Univ. Math. J.
**21**, 951–960 (1972)MathSciNetCrossRefzbMATHGoogle Scholar