Introduction to Toeplitz operators I just finished my undergraduate education in mathematics, and i'm starting a graduate program, and i get interest for learning to work with Toeplitz operators, but i have no background with functional analysis yet, so i was wondering if you could help me to indicate me a route of books or other references to learn about Toeplitz operators beginning with functional analysis. Thank you in advance.
 A: You may want to spend some time looking into the history of this subject, and to see how this Math was originally used and how it evolved. Most of the subject has its roots in convolution integral equations on the real line, and the important techniques of Complex Analysis to solve such problems through factoring holomorphic functions originated with Norbert Wiener.
Wiener was looking at filtering and prediction, including radar, and was having trouble knowing how to solve certain convolution integral equations because these equations seemed to require information in the negative time direction. This had people stumped. Wiener used the Fourier/Laplace transform to obtain a new equation, and he split everything into positive and negative times, which split things into holomorphic functions in the upper-half plane and holomorphic functions in the lower-half planes. If a function had a log-summable modulus, Wiener found that he could factor the function into a product of one function which was holomorphic in the upper half-plane and another which was holomorphic in the lower half-plane. He was then able to rearrange the terms into an equation where the pieces on one side were holomorphic in one half-plane, and the pieces on the other side were holomorphic in the other half-plane. Now he had an entire function which he could either show was bounded or of polynomial growth. Suddenly, the equations split because the expression was a constant, which meant both sides were constant! Amazingly, the indeterminate half of the time equation was gone. Violla! Filtering and prediction. Most of this remained classified until after WWII.
Wiener looked at winding numbers of symbols to determine the polynomial character, and he related this to Fredholm index, which one would expect. The projections of functions back into holomorphic functions, the factoring theorems, and the Toeplitz formulations were strongly developed in this context. Wiener and Paley developed their Wiener-Paley theorem characterizing the square integrable holomorphic functions on a half-plane, and everything suddenly became $L^{2}$ complex function theory on a half-plane. Later people moved the theory onto the unit disk where things where studied using Hardy spaces. But the genesis of the relevant ideas is found in Wiener's original work, which I think you'll find readable and enjoyable.
I suggest you browse the work of Wiener and Paley and see how this started in the half-plane, instead of the disk. Very instructive. Wiener and Paley wrote a very readable, classic book "Fourier Transforms in the Complex Domain." Volume III of Wiener's collected works has many interesting articles, especially concerning the Fredholm index / winding number connection.
A great introduction to Spectral Theorem in general, with a nice chapter on index theory and such symbols can be found in a book by Arveson, "A Short Course on Spectral Theory". This is short book--about 125 pages that is abstract, elegant and compact, and focuses on Operator Algebras. This book is written for a well-prepared graduate student, and may be a bit much right now. But keep this one in mind for later.
A: The book "Banach Algebra Techniques in Operator Theory" written by Ron Douglas, has an excellent chapter on Toeplitz operators at the end of the book. It's where I started when I began my PhD research. He also has a follow up book "Banach Algebra Techniques in the theory of Toeplitz Operators."
You will need some background in working with Hardy Spaces. Douglas's book has a good intro to this, but a better book for a beginner (and cheaper) is Banach Spaces of Analytic Functions by Hoffman. (This is free on the internet archive, archive.org)
Finally there is Analysis of Toeplitz Operators by Bottcher and Silbermann. If your university pays for a subscription with Springer, you might be able to get it for free through Springerlink. That's what I did.
