Operator Approximant Problems Arising from Quantum Theory

verfasst von: Philip J. Maher

Verlag: Springer International Publishing

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This book offers an account of a number of aspects of operator theory, mainly developed since the 1980s, whose problems have their roots in quantum theory. The research presented is in non-commutative operator approximation theory or, to use Halmos' terminology, in operator approximants. Focusing on the concept of approximants, this self-contained book is suitable for graduate courses.

Inhaltsverzeichnis

Frontmatter

Chapter 1. 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.

Philip J. Maher

Chapter 2. Preliminaries

Abstract

This chapter presents, and highlights, material (much of which will be familiar to the reader) required for the rest of this work.

Philip J. Maher

Chapter 3. Self-Adjoint and Positive Approximants

Abstract

The subject of operator approximation dates back to the 1950^s to the seminal work of Fan and Hoffman [16] who studied, in part, self–adjoint approximation. It was not, however, until the 1970^s that the subject seems to have taken of, precipitated by the papers of Halmos [21, 22]. In [21] Halmos coined the term “approximant” and focused mainly on positive approximation.

Philip J. Maher

Chapter 4. Commutator Approximants

Abstract

We study approximation by commutators AX − XA, by generalized commutators AX − XB and by self–commutators X ^∗ X − XX ^∗ for varying X in the context of \(\mathcal{L}(H)\) and \(\mathcal{C}_{p}\).

Philip J. Maher

Chapter 5. Spectral, and Numerical Range, Approximants

Abstract

The theory of spectral approximants presents a precise geometric way of specifying approximants. The theory was initiated by Halmos [22] and later extended to the context of \(\mathcal{C}_{p}\) by Bouldin [11] and Bhatia [8]. More recently the related concept of numerical range approximant was introduced [25].

Philip J. Maher

Chapter 6. Unitary, Isometric and Partially Isometric Approximation of Positive Operators

Abstract

This chapter is about approximation of positive operators by operators that in some sense preserve size: by—in ascending order of generality—unitaries, isometries and partial isometries.

Philip J. Maher

Backmatter

Titel: Operator Approximant Problems Arising from Quantum Theory
verfasst von: Philip J. Maher
Verlag: Springer International Publishing
Electronic ISBN: 978-3-319-61170-9
Print ISBN: 978-3-319-61169-3
DOI: https://doi.org/10.1007/978-3-319-61170-9