The books 1) and 2) are very comprehensive and treat the classical spectral theory of differential operators in great detail. Yoshida's book is, however, far less systematic and goes back to the basic principles of topology, analysis, and measure theory. It is therefore a very good introduction in the field as a whole but will only pay for those readers who read it in large parts and learn how to find the relevant items in the book. Book 2) is far more systematic and provides an excellent introduction into topological vector space and operator theory, both on Banach and Hilbert spaces. Again, because of its sheer volume one has to read big parts before profiting from it, only that a selective reading in this case is much easier.
This book is an introduction to the basic facts about functional analysis and operator theory, written by two mathematicians which are not analysts, though. The big advantage is that you collect a lot of valuable knowledge in a relatively short time, the drawback - besides being in German - is that you have to watch out for small mistakes (which are never really relevant, though) - a very good training!
This book is a very good introduction to the theory of the locally convex topological vector spaces, a huge family of spaces of which Banach and Hilbert spaces form only the "tip of the iceberg". All fundamental theorems are presented in great generality but with carefully chosen concepts, quite a few examples, and a very decent pace of arguing. Everybody should read at least chapters 1, 2, and 4.
These two books provide an extensive treatment of the spectral theorem in (I, Ch. 7 and Ch. 8) together with a wealth of examples and detailed discussions in IV. Both books provide entertaining and sometimes beautiful material with the danger, though, to drown in an ocean of details which provides sometimes little structure.
This is a relatively small sized book which is very relevant for our lecture and will play a role occasionally even though manifolds and vector bundles are not mentioned at all. However, the proofs are not given in great detail, the author demands a lot from the reader's attentiveness and cleverness. However, succeeding with this book is certainly an accomplishment.
This book works more on the Banach space side and less with Hilbert spaces, but it gives a careful and enjoyable introduction into Fredholm theory and, in particular, to one of its main applications, the so-called Toeplitz operators. Nevertheless, this book is useful but not in the main focus of the lecture.
This book deals with the geometric meaning of the eigenvalues of the so-called Laplace-Beltrami operator on closed Riemannian manifolds. It is quite relevant for the latter part of the lecture and provides many interesting aspects for further study.
This book is a classical treatment of the Laplace-Beltrami operator and its spectrum, preceeding the previously mentioned book, and at the same time it provides a short but excellent introduction into the main principles of Riemannian geometry. The book focuses on isospectral Riemannian manifolds that is (closed) Riemannian manifolds with the same spectrum of their Laplace-Beltrami operators which are, however, not isometric. We will touch upon this topic only briefly, also in the seminar, but otherwise the same remarks apply as for item 7.
These two volumes are a "bible" for large parts of the theory we have treated in the lecture, with special emphasis on ordinary differential operators - and with lots of very good exercises.
This book presents one of the first complete treatments of the spectral theorem but is particularly important through its first chapters which give a natural and thorough introduction to the theory of real functions which today is largely replaced by measure theory, with a slightly different point of view.