Rigorous Definition of Infinitesimal Projection Operator?

[Cruikshank]

I've been reading Thomas Jordan's Linear Operators for Quantum Mechanics, and I am stalled out at the bottom of page 40. He has just defined the projection operator E(x) by E(x)(f(y)) = {f(y) if y≤x, or 0 if y>x.} Then he defines dE(x) as E(x)-E(x-ε) for ε>0 but smaller than the gap between eigenvalues. Okay, so long as the eigenvalue spectrum is discrete...but then he just announces that ∫dE= E (limits -∞ to ∞) when it is supposed to be a discrete sum. I can sort of almost accept the handwave there, but then in the next line, he states ∫xdE = A, and that just doesn't make any sense to me. Sure, in a vague handwaving sense, maybe, but how does one define this so that it rigorously makes sense for operators with continuous eigenvalue spectrum?

Is there a particular reference you recommend? Please, not Von Neumann--I spent months on that book and found over 130 errata; it was the worst way imaginable to learn this math.

Relatedly, I've looked at several texts on the theory of integration (I keep running into "Stiejles" integrals references and Lesbegue integration, but haven't found a good book yet for learning the details that didn't make it more boring than doing taxes.) Any suggestions there? Thanks.

 

[Fredrik]

I don't think the relevant theorems can be proved to a typical QM student in less than 300 pages. 500 pages sounds about right. I still haven't made it to the end. The topics you need to study are point-set topology, integration theory, introductory functional analysis (the basics of Hilbert spaces), operator algebras and spectral theorems. And you probably need to study linear algebra again.

I find it difficult to recommend books for you, because I chose a path that probably isn't the best one. I used Axler's "Linear algebra done right" to refresh my memory and learn a few new things. That was a good choice. Axler probably has the best selection of topics for a QM student. Then I dove right into Conway's "A course in functional analysis", but I found it impossibly hard. You have to know point-set topology really well to give that one a shot, and I didn't at the time. (In particular, you have to know everything about compact sets very well). So I moved on to Sunder's "Functional analysis: Spectral theory", and started with the appendices on linear algebra and topology. I also used some other books to supplement that appendix on topology, in particular Munkres. This was an OK way to learn topology, but there are probably better ways. When I had learned some topology, I found the early chapters of Conway quite useful (e.g. the stuff about the projection theorem and orthonormal bases), even though they were still hard to read. The proofs in Friedman's "Foundations of modern analysis" were much easier to follow.

I also used Friedman to learn some stuff about integration theory, but I found the presentation very difficult to follow. I supplemented it with Lang's "Real and functional analysis", which has a similar approach, and Capinski & Kopp's "Measure, integral and probability", which has a more traditional approach. I figured it out eventually, but I had to work very hard to get there.

I got distracted by other things a few years ago, and had to put this aside, but I recently started working on it again. Right now I'm studying sections of Murphy's "C*-algebras and operator theory" and Sunder's book again.

People have told me that Kreyszig's book on functional analysis is very good. It's too late for me to switch to that approach, but It may be a good place to start for you.

 

[micromass]

Conway has a new book by the way: https://www.amazon.com/dp/0821890832/?tag=pfamazon01-20 It's easier and a lot better than his functional analysis book.
He also has a truly excellent book on operator algebras, but Murphy is also really nice.