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Published in:
2017 | OriginalPaper | Chapter
1. What This Book Is About: Approximants
Author : Philip J. Maher
Published in: Operator Approximant Problems Arising from Quantum Theory
Publisher: Springer International Publishing
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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.