Noncommutative Analysis

Month: October, 2012

Advanced Analysis, Notes 5: Hilbert spaces (application: Von Neumann’s mean ergodic theorem)

In this lecture we give an application of elementary operators-on-Hilbert-space theory, by proving von Neumann’s mean ergodic theorem. See also this treatment by Terry Tao on his blog.

For today’s lecture we will require the following simple fact which I forgot to mention in the previous one.

Exercise A: Let A, B \in B(H). Then \|AB\| \leq \|A\| \|B\|.

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Advanced Analysis, Notes 4: Hilbert spaces (bounded operators, Riesz Theorem, adjoint)

Up to this point we studied Hilbert spaces as they sat there and did nothing. But the central subject in the study of Hilbert spaces is the theory of the operators that act on them. Paul Halmos, in his classic paper “Ten Problem in Hilbert Space“, wrote:

Nobody, except topologists, is interested in problems about Hilbert space; the people who work in Hilbert space are interested in problems about operators.

Of course, Halmos was exaggerating; topologists don’t really care much for Hilbert spaces for their own sake, and functional analysts have much more to say about the structure theory of Hilbert space then what we have learned. Nevertheless, this quote is very close to the truth. We proceed to study operators.  Read the rest of this entry »

Advanced Analysis, Notes 3: Hilbert spaces (application: Fourier series)

Consider the cube K := [0,1]^k \subset \mathbb{R}^k. Let f be a function defined on K.  For every n \in \mathbb{Z}^k, the nth Fourier coefficient of f is defined to be

\hat{f}(n) = \int_{K} f(x) e^{-2 \pi i n \cdot x} dx ,

where for n = (n_1, \ldots, n_k) and x = (x_1, \ldots, x_k) \in K we denote n \cdot x = n_1 x_1 + \ldots n_k x_k.  The sum

\sum_{n \in \mathbb{Z}^k} \hat{f}(n) e^{2 \pi i n \cdot x}

is called the Fourier series of f. The basic problem in Fourier analysis is whether one can reconstruct f from its Fourier coefficients, and in particular, under what conditions, and in what sense, does the Fourier series of f converge to f.

One week into the course, we are ready to start applying the structure theory of Hilbert spaces that we developed in the previous two lectures, together with the Stone-Weierstrass Theorem we proved in the introduction, to obtain easily some results in Fourier series.

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Advanced Analysis, Notes 2: Hilbert spaces (orthogonality, projection, orthonormal bases)

(Quick announcement: all lectures will from now on take place in room 201). 

In the previous lecture, we learned the very basics of Hilbert space theory. In this lecture we shall go one little bit further, and prove the basic structure theorems for Hilbert spaces.

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Reflections on the New York Journal of Mathematics

As I have just announced in a previous post, Matt Kennedy and I have just published a paper in the New York Journal of Mathematics.

The New York Journal of Mathematics is a nonprofit electronic journal, which posts papers openly online so that anyone can read them without any subscription fee. And of course (funny that this has to be noted) it does not require that authors pay for having their papers published. It exists simply for the benefit of mathematical research and the mathematical community. This is how journals should be. There are others like it: there is the BJMA in which I have published in once. See also the list of free online math journals here.

The NYJM is more than a community project – it is a good general math journal. How do I know? The same way I know that other good journals are good: first, I take a look at the editorial board, and I see that there are distinguished mathematicians among the editors (and most importantly for me, I check that there is an editor who is close to my field so he/she will know what to do with my submission); second, I check to see if mathematicians whom I know and highly respect have published there; third, just to be on the safe side, I can browse the index and see if any famous mathematicians which I have heard of have published there too; fourth, I check to see if the journal is on MathSciNet’s Citation Database Reference List (it is); after that I may or may not decide to submit (and this of course also depends on what my coauthor thinks), and if I submit I also get an impression of how professional, smooth and fast the publishing process is. My impression from my recent experience is that the publishing process in NYJM is as professional, smooth and fast as I could hope for.

Unfortunately, some committees which make decisions regarding tenure and promotion also need to decide if the journals in which candidates publish are good journals. There are several “bibliometrical” tools which help committees and administrators figure out if journals are any good. Here at BGU the tool usually used is something called ISI Web of Knowledge. Now guess what ISIWoK says about NYJM. Seriously, guess: do you think that ISIWoK says that NYJM is a good journal or an OK journal or a bad journal?

HA! Trick question! According to ISIWoK, the New York Journal of Math doesn’t exist. There is no such journal. Now, the NYJM has been coming out since 1994, so somebody at ISIWoK hasn’t been doing a very good job. Or maybe they have?

Well: luckily my university has decided to treat NYJM as a real journal (I am sorry to admit that I probably would not have published there otherwise). Unfortunately, there is still a way to go: my university still uses ISIWoK to count citations, so for this paper of mine there will be no data. I hope that this will change before I am up for promotion.


UPDATE February 5, 2013: Mark Steinberger commented below that NYJM is now covered by Thomson-Reuters Web of Science, and that this is retroactive to Volume 16 (2010).