This book serves as a fairly comprehensive review of results in spectral theory from the past decade. It is remarkably detailed and has a great number of references (271 to be precise). The content is somewhat advanced and relies on the reader's previous exposure to basic Hilbert and Banach space theory. That being said, there is a quick review of the basics in Chapter 1. There are no exercises within this book, so the main use I would say is a reference text for a mathematician actively working in the areas of analysis and partial differential equations. Perhaps this book could also be used at the graduate level after students have taken a first course in functional analysis.

Speaking of Chapter 1...it is not merely the anticipated basic review of classical functional analysis results (e.g. Hahn-Banach Theorem, Closed Graph Theorem, etc.). Rather, after a swift introduction to compact operators, the authors delve right into Fredholm maps and measures of non-compactness. Right away it is impressed upon the reader that it may be necessary to briefly review an introductory functional analysis text to freshen up. It is certainly quite novel to have the non-standard topic of measures of non-compactness so early in the book! As someone who is decently competent in the language of functional analysis, I found this extremely interesting and instructive. Chapter 2 is a continuation of non-standard functional analysis results: entropy numbers and s-numbers.

The first 4 chapters contain most of the abstract theory. Chapter 3 contains some applications of the theory to general second order \emph{quasi-differential} equations. Such equations are of the form \( -(p\phi')' +r\phi' +q\phi' = \lambda w \phi, \quad \lambda \in \mathbb{C}\)

where \( p,w\) are real valued and \( r,q, \) are complex valued. This equation is not a second order differential equation in the usual sense since \( \phi''\) need not exist even almost everywhere! The treatment of such equations in Chapter 3 is quite thorough. In Chapter 4 the authors specialize a bit to Hilbert spaces, with a particular focus on sectorial operators.

Chapter 5 provides a crash course in Sobolev space theory. This treatment is still fairly comprehensive even though it is only one chapter in the book. This chapter also provides a concrete setting for applying some of the previously developed abstract results.

The rest of the book is devoted to applications of the theory to second order elliptic differential operators. A large focus is on the essential spectra of such differential operators (in fact, Chapter 9 is devoted entirely to essential spectra!). Particular operators of interest to the authors are the Dirichlet and Neumann Laplacian (both on bounded domains and unbounded domains) as well as Schrödinger operators \( -\Delta + q \) for both real and complex-valued \( q\).

Overall, I am astounded by the staggeringly detailed nature of this book. It is one of the few books out there that presents such technical material in such a thorough, complete way. I highly recommend this book as a reference for mathematicians interested in functional analysis and its applications to differential equations.

Eric Strachura is currently an Assistant Professor of Mathematics at Kennesaw State University. He is generally interested in analysis and partial differential equations, especially treated with functional analysis techniques.