Alternatives to Reed & Simon's book

Question

I'm looking for alternative references on functional analysis which can replace Reed & Simon's book. I'm a big fan of Reed & Simon, but I'd like to have other perspectives on some of its contents. One of great advantages of Reed & Simon is their vast number of applications, specially to physics, and I think this might be difficult to find, so you can consider applications to physics a "plus" but not essential. Topics I'm more interested in include: general theory of bounded and unbounded operators on Hilbert spaces, spectral theory for compact and self-adjoint operators (in particular, the functional calculus and Borel funcional calculus) and operator topologies (e.g. weak, weak* and strong operator topologies). Modern expositions are specially welcomed; I know, for example, that a more general approach to the functional calculus and Borel functional calculus can be given by means of $C^{*}$-algebras, where the operator algebras become just particular cases [this is not the approach on Reed & Simon, however].

Note: I know that many functional analysis books cover the listed topics, but my purpose is to find something compatible with Reed & Simon in terms of language, notation and structure. For instance, their approach is really self-contained and focused. Moreover, they seem to have a lot of results which are not only advanced material but also useful for applications in physics.

Answer 1

Barry Simon was one of the leading experts in research on mathematical quantum mechanics, more precisely the operator theoretic side of it, and this immense expertise went into his books with Reed. This is just to say that you are setting a high bar if you want a book that includes advanced material with useful applications to physics.

If there is one researcher whose achievements in that period rival Simon's, it is probably Tosio Kato, and incidentally, he too wrote a book, Perturbation Theory for Linear Operators. Now I don't think I have come across someone who has used this book to study the basics of functional analysis, although it does contain a lot of the classical material of a first course in functional analysis. But since this does not seem to be the focus here, it might be just what you are looking for (and it actually has a broader range than the title suggests). As far as I remember, it does not go too much into the physical applications, but you will still find a lot of useful material if you want to study, say, Schrödinger operators.

Answer 2

Specifically for spectral theory, you might be interested in "Quantum Theory for Mathematicians" by Brian C. Hall. The textbook covers quantum mechanics from a mathematical perspective, and it builds all of the spectral theory along the way. It comes equipped with proofs of various versions of the spectral theorems for both bounded and unbounded operators. It also contains the background on bounded and unbounded operators.

This book is more built around physics than Reed and Simon's text, and I wouldn't call it a functional analysis textbook. It is much more of along the lines of "let's carefully build the math that we need to do the physics," rather than "we're doing math, and here are some applications in physics." That might still fit within the scope of what you're looking for, and it's a very well-written book. It contains much more than just what I mentioned, as well (including background on Lie groups, Lie algebras, and group representations).

Also, it is a great reference for the aforementioned spectral theory.