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What This Book Is About: Approximants

  • Philip J. Maher
Chapter

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. 1.
    J.G. Aiken, J.A. Erdos, J.A. Goldstein, Unitary approximation of positive operators. Ill. J. Math. 24, 61–72 (1980)MathSciNetzbMATHGoogle Scholar
  2. 21.
    P.R. Halmos, Positive approximants of operators. Indiana Univ. Math. J. 21, 951–960 (1972)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Philip J. Maher
    • 1
  1. 1.LondonUK

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