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Advanced Analysis, Notes 11: Banach spaces (weak topologies, Alaoglu’s theorem)
Let be the Banach space of continuous functions on the interval with the sup norm. Consider the following sequence of functions defind as follows. and for all , is equal to zero on the interval...
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Advanced Analysis, announcement
I have been writing my own lecture notes (based, of course, on an array of well known references and also on some of my notes from my graduate studies) for the course Advanced Analysis. I have decided...
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Advanced Analysis, Notes 12: Banach spaces (application: existence of Haar...
This post is dedicated to my number one follower for her fourteenth birthday,which I spoiled… Let be a compact abelian group. By this we mean that is at once both an abelian group and a compact...
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Advanced Analysis, Notes 13: Banach spaces (convex hulls and the Krein–Milman...
It would be strange to disappear for a week without explanations. This blog was not working for the past week because of the situation in Israel. The dedication in the beginning of the previous post...
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Advanced Analysis, Notes 14: Banach spaces (application: the...
In this post we will use the Krein–Milman theorem together with the Hahn–Banach theorem to give another proof of the Stone–Weierstrass theorem. The proof we present does not make use of the classical...
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The remarkable Hilbert space H^2 (Part I – definition and interpolation theory)
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...
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The remarkable Hilbert space H^2 (Part II – multivariable operator theory and...
This post is the second post in the series of posts on the d–shift space, a.k.a. the Drury–Arveson space, a.k.a. (see this previous post about the space ). 1. Model theory One of the ways in which one...
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Matthew Kennedy awarded CMS 2012 Doctoral Prize
One of the nicest things that happen on math blogs is that people write expository posts on other people’s work. Tim Gowers and Terry Tao have set a fine example in their expositions of the works of...
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The remarkable Hilbert space H^2 (part III – three open problems)
This is the last in the series of three posts on the d–shift space, which accompany/replace the colloquium talk I was supposed to give. The first two parts are available here and here. In this post I...
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Advanced Analysis, Notes 15: C*-algebras (square root)
This post contains some make–up material for the course Advanced Analysis. It is a theorem about the positive square root of a positive element in a C*-algebra which does not appear in the text book we...
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Worse than Elsevier, worse than …
I recently received the following email from Cambridge University Press: Submit your article online now Dear Dr Shalit We are delighted to announce that the online submission systems forForum of...
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Happy new year!
It is good to start the new year with a positive headline in my blog. So to any occasional visitor of this blog who comes from afar: if this is your new year, then I wish you a happy new year! I will...
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Souvenirs from Bangalore
I recently returned from the two week long workshop and conference Recent Advances in Operator Theory and Operator Algebras which took place in ISI Bangalore. As I promised myself before going, I was...
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Advanced Analysis, Notes 16: Hilbert function spaces (basics)
In the final week of the semester we will study Hilbert function spaces (also known as reproducing kernel Hilbert spaces) with the goal of presenting an operator theoretic proof of the classical Pick...
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Advanced Analysis, Notes 17: Hilbert function spaces (Pick’s interpolation...
In this final lecture we will give a proof of Pick’s interpolation theorem that is based on operator theory. Theorem 1 (Pick’s interpolation theorem): Let , and be given. There exists a function...
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Some links and announcements
The course “Advanced Analysis” is over. The lecture notes (the part that I prepared) are available here. Comments are very welcome. I hope to teach this course again in the not too far future and...
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Partial results
What is more wonderful: something wonderful, or the moment before the wonderful something happens? Recently I proved a result that can best be described as a partial result. This is certainly not what...
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Mathematics on mathematics
[Dedicated to my best friend (my best friend!)] This post is the outline of a talk (or perhaps the talk is an outline of this post) that I will give on February 28 on our “open day” to prospective...
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Forty five years later, a major open problem in operator algebras is solved
A couple of days ago, Ken Davidson and Matt Kennedy posted a preprint on the arxiv, “The Choquet boundary of an operator system“. In this paper they solve a major open problem in operator algebras,...
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A sneaky proof of the maximum modulus principle
The April 2013 issue of the American Mathematical Monthly has just appeared, and with it my small note “A Sneaky Proof of the Maximum Modulus Principle”. Here is a link to the current issue on the...
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