[Math] Reference text for Hilbert space theory.

Question

I am searching for a reference that contains a detailed discussion of most of the topics in Hilbert space theory. I am both interested in the geometry of Hilbert spaces and operators on Hilbert spaces.

I am familiar with several excellent texts on Banach space theory; for example, Megginson's An Introduction to Banach Space Theory and Albiac & Fanton's Topics in Banach Space Theory. However, I am not aware of similar types of books for the theory of Hilbert spaces.

The book that comes most closely to what I have in mind is probably Halmos' A Hilbert Space Problem Book. However, as the title of this book indicates, this book is meant as a problem book and not really a reference text.

I am familiar with general topology, abstract measure theory, and functional analysis; so it is no problem if the book has these topics as a prerequisite (as Halmos' book has).

All suggestions and comments are welcome.

Answer 1

(Caveat: while these all look promising to me, I haven't read any of them myself.)

• William Arveson, A Short Course on Spectral Theory (Springer 2002)
• Bela Bollobas, Linear Analysis (2nd ed. Cambridge University Press 1999)
• Ronald G. Douglas, Banach Algebra Techniques in Operator Theory (Academic Press 1972 - there's a 2nd ed. from Springer, 1998, apparently not much changed from the 1st ed.)
• Gilbert Helmberg, Introduction to Spectral Theory in Hilbert Space (North-Holland, Amsterdam 1969; corr. 2nd pr. 1975; repr. Dover 2008(?))
• N. Young, An Introduction to Hilbert Space (Cambridge University Press 1988)