The author of Linear Operators and their Spectra, E. Brian Davies, who, together with Barry Simon, recently carried out some dramatic work on pseudo-spectra, describes the book under review as pitched “halfway between a textbook and a monograph” and directs it at “graduate students and young researchers.” For what it’s worth, coming from an interested outsider like me, I’d put the book about three-fourths of the way between a text and a monograph: it’s exactly on target for some one who has already made a strong start in functional analysis, beyond a first course in graduate school, and who is sufficiently mathematically sophisticated to wish to pursue non-self-adjoint operators.
Standard functional analysis, for lack of a better word, is of course largely concerned with self-adjoint operators on Hilbert spaces, betraying its co-evolution with certain famous trends in quantum mechanics For instance, it is no exaggeration to say that such a seminal contribution as John von Neumann’s The Mathematical Foundations of Quantum Mechanics qualifies as both a landmark of mathematical physics and a definitive work in the early history of functional analysis.
It is in this connection that the spectral theorem should be mentioned. Davies notes on p. 143 of his book that “the general form of the theorem was obtained in dependently by Stone and von Neumann between 1929 and 1932.” Then he goes on to say that while “[i]t is undoubtedly the most important result in the subject [and] [t]he theorem is used in several places in the book… we do not give a proof.” The reader is referred to Dunford and Schwartz.
The point to be taken is that Davies’ has bigger (or different) fish to fry, to wit, pseudo-spectra, introduced in chapter 9. Here is a revealing passage in this connection (p. 245): “…pseudo-spectra arose as a result of the realization that several ‘pathological’ properties of highly non-self-adjoint operators were closely related … [e.g.] the existence of appropriate eigenvalues away from the spectrum; the instability of the spectrum under small perturbations; the anomalous response of systems subject to a periodic driving term [and other themes].” Manifestly, very interesting things are going on, providing a wonderful context in which to learn some avant garde operator theory, and, for those called to do so, to start to work in this field.
The first half-dozen chapters of Linear Operators and their Spectra present a nice treatment of functional analysis’ more familiar themes, including Lp spaces, a good dosage of Fourier analysis, “intermediate operator theory,” operators on Hilbert space, and one-parameter semigroups. The rubber hits the road in the latter half of the book, where the focus falls on such topics as perturbation theory, Markov chains, ergodicity, and non-self-adjoint Schrödinger operators. The book also contains examples and problems to aid in this good cause of spreading the word about some interesting modern analysis with a very distinguished pedigree.
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles.
Preface; 1. Elementary operator theory; 2. Function spaces; 3. Fourier transforms and bases; 4. Intermediate operator theory; 5. Operators on Hilbert space; 6. One-parameter semigroups; 7. Special classes of semigroup; 8. Resolvents and generators; 9. Quantitative bounds on operators; 10. Quantitative bounds on semigroups; 11. Perturbation theory; 12. Markov chains and graphs; 13. Positive semigroups; 14. NSA Schrödinger operators.