Category Archives: Expository

The remarkable Hilbert space H^2 (Part I – definition and interpolation theory)

d shift space, Expository, Operator theory, Talks,

This series of posts is based on the colloquium talk that I was supposed to give on November 20, at our department. As fate had it, that week studies were cancelled.

Several people in our department thought that it would be a nice idea if alongside the usual colloquium talks given by invited speakers which highlight their recent achievements, we would also have some talks by department members that will be more of an exposition to the fields they work in. So my talk was supposed to be an exposition to the setting in which much of the research I do goes on.

The topic of the “talk”  is the Hilbert space $latex H^2_d$. There will be three parts to this series:

  1. Definition and interpolation theory. 
  2. Multivariate operator theory and model theory
  3. Current research problems

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William Arveson

Arveson, Expository, Functional analysis, Operator theory,

William B. Arveson was born in 1934 and died last year on November 15, 2011. He was my mathematical hero; his written mathematics has influenced me more than anybody else’s. Of course, he has been much more than just my hero, his work has had deep and wide influence on the entire operator theory and operator algebras communities. Let me quickly give an example that everyone can appreciate: Arveson proved what may be considered as the “Hahn-Banach Theorem” appropriate for operator algebras. He did much more than that, and I will expand below on some of his early contributions, but I want to say something before that on what he was to me.

When I was a PhD student I worked in noncommutative dynamics. Briefly, this is the study of actions of (one-parameter) semigroups of *-endomorphisms on von Neumann algebras (in short E-semigroups). The definitive book on this subject is Arveson’s monograph “Noncommutative Dynamics and E-Semigroups”. So, naturally, I would carry this book around with me, and I would read it forwards and backwards. The wonderful thing about this book was that it made me feel as if all my dreams have come true! I mean my dreams about mathematics: as a graduate student you dream of working on something grand, something important, something beautiful, something elegant, brilliant and deep. You want your problem to be a focal point where different ideas, different fields, different techniques, in short, all things, meet.

When reading Arveson there was no doubt in my heart that, e.g., the problem classifying E-semigroups of type I was a grand problem. And I was blown away by the fact that the solution was so beautiful. He introduced product systems with such elegance and completeness that one would think that this subject has been studied for the last 50 years. These product systems were measurable bundles of operator spaces – which turn out to be Hilbert spaces! – that have a group like structure with respect to tensor multiplication. And they turn out to be complete invariants of E-semigroups on $latex B(H)$. The theory set down used ideas and techniques from Hilbert space theory, operator space theory, C*-algebras, group representation theory, measure theory, functional equations, and many new ideas – what more could you ask for? Well, you could ask that the new theory also contribute to the solution of the original problem.

It turned out that the introduction of product systems immensely advanced the understanding of E-semigroups, and in particular it led to the full classification of type I ones.

So Arveson became my hero because he has made my dreams come true. And more than once: when reading another book by him, or one of his great papers, I always had a very strong feeling: this is what I want to do. And when I felt that I gave a certain problem all I thought I had in me, and decided to move on to a new problem, it happened that he was waiting for me there too.

I wish to bring here below a little piece that I wrote after he passed away, which explains from my point of view what was one of his greatest ideas.

For a (by far) more authoritative and complete review of Arveson’s contributions, see the two recent surveys by Davidson (link) and Izumi (link).

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Functional Analysis – Introduction. Part I

Advanced Analysis, 20125401, Expository, Functional analysis, Operator theory,

I begin by making clear a certain point. Functional analysis is an enormous branch of mathematics, so big that it does not seem appropriate to call it “a branch”, it sometimes looks more like another tree. When I will talk below about functional analysis, I will mean “textbook functional analysis” and not “research functional analysis”. By this I mean that I will only refer to the core of the theory which is several decades old and which is more-or-less agreed to be the essential and basic part of the subject.

The goal of this post is to serve as an introduction to the course “Advanced Analysis, 201.2.5401”, which is a basic graduate course on (textbook) functional analysis. In the lectures I will only have time to give a limited description of the roots of the subject and the motivation will have to be brief. Here I will aim to describe what was the climate in which this tree grew, where are its roots and what are its fruits.

To prepare this introduction I am relying on the following sources. First and foremost, my love of the subject and my point of view on it were strongly shaped by my teachers, and in particular by Boris Paneah (my Master’s thesis advisor) and Baruch Solel (my PhD. thesis advisor). Second, I learned a lot on the subject from the book “Mathematical Thought from Ancient to Modern Times” by M. Kline and from the notes sections of Rudin’s and Reed-Simon’s books “Functional Analysis”.

And a warning to the kids: this is a blog, not a book, and if you really want to learn something go read the books (the books I mentioned have precise references).

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Functional analysis – a preface to the introduction

Advanced Analysis, 20125401, Expository, Thoughts on mathematics,

I am planning to write a post that will be an introduction to the course “Advanced Analysis”, which I shall be teaching in the fall term. The introduction is to comprise two main themes: motivation and history. I was a little surprised to find out – as I was preparing the introduction – that, looking from the eyes of a student, the history of subject provided little motivation. I also began to oscillate between two opposite (and equally silly) viewpoints. The first viewpoint is that functional analysis is a big and respectable field of mathematics, which needs no introduction; let us start with the subject matter immediately since there is so much to learn. The second viewpoint is that there is absolutely no point in studying (or teaching) a mathematical theory without understanding its context and roots, or without knowing how it applies to problems outside of the theory’s borders. Pondering these, I found that I had some things to say before the introduction, which may justify the introduction or give it the place I intend.

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