Category Archives: Complex variables

Advanced Analysis, Notes 19: The holomorphic functional calculus II (definition and basic properties)

In this post we continue our discussion of the holomorphic functional calculus for elements of a Banach algebra (or operators). The beginning of this discussion can be found in Notes 18. Continue reading Advanced Analysis, Notes 19: The holomorphic functional calculus II (definition and basic properties)

Spectral sets and distinguished varieties in the symmetrized bidisc

In this post I will write about a new paper, “Spectral sets and distinguished varieties in the symmetrized bidisc“, that Sourav Pal and I posted on the arxiv, and give the background to understand what we do in that paper.

Continue reading Spectral sets and distinguished varieties in the symmetrized bidisc

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 journal’s website, and here is a link to a version of the paper on my homepage. As the title suggest, the note contains a new proof — which I find extremely cool — for the maximum modulus principle from the theory of complex variables. The cool part is that the proof is based on some basic linear algebra. The note is short and very easy, and I am not going to say anything more about the proof, except that it relates to some of my “real” research (the way in which it relates can be understood by reading the Note and its references).

I am writing this post not only to publicize this note, but also to record somewhere my explanation why I have been behaving in a sneaky fashion. Indeed, this is the first paper that I wrote which I did not post on the Arxiv. Why?

Unlike research journals, the American Mathematical Monthly is a journal which has, if I am not mistaken, actual subscribers. I mean real people, some of them perhaps old school (like myself), and I could see them waiting to receive their copy in the mailbox, and then when the new issue finally arrives they gently open the envelope — or perhaps they tear it open, depending on their custom — after which they sit down and browse through the fresh issue. I could believe that there are such persons (for I myself am such a person) that do not look at the online version of the journal even though they have access, because that would spoil their fun with the paper copy which is to arrive a few days later.

Now I wouldn’t like to spoil a small pleasure of a subscriber, somewhere out there. So I did not post the Note on the Arxiv, lest it pop up on somebody’s mailing list. “Oh, this I have already seen…”. I shall not be resposible for such spoilers! So I decided to keep my note relatively secret, putting it on my homepage, but putting off the Arxiv until the journal really gets published and all the physical copies are safely in the mailboxes of all subscribers. I made this decision about a year ago from now, and to tell the truth I felt that a year is a terribly long time to wait. In the end, this year appears much much shorter from this end than from the other one.

(I guess that it does not matter much if I put it on the Arxiv now: in the meanwhile I discovered that google scholar has managed to figure out that such a note exists on somebody’s webpage. Probably I will post it on the Arxiv, for the sake of all things being in good order).

Advanced Analysis, Notes 17: Hilbert function spaces (Pick’s interpolation theorem)

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 $latex z_1, ldots, z_n in D$, and $latex w_1, ldots, w_n in mathbb{C}$ be given. There exists a function $latex f in H^infty(D)$ satisfying $latex |f|_infty leq 1$ and 

$latex f(z_i) = w_i ,, ,, i=1, ldots, n$

if and only if the following matrix inequality holds:

$latex big(frac{1-w_i overline{w_j}}{1 – z_i overline{z_j}} big)_{i,j=1}^n geq 0 .$

Note that the matrix element $latex frac{1-w_ioverline{w_j}}{1-z_ioverline{z_j}}$ appearing in the theorem is equal to $latex (1-w_i overline{w_j})k(z_i,z_j)$, where $latex k(z,w) = frac{1}{1-z overline{w}}$ is the reproducing kernel for the Hardy space $latex H^2$ (this kernel is called the Szego kernel). Given $latex z_1, ldots, z_n, w_1, ldots, w_n$, the matrix

$latex big((1-w_i overline{w_j})k(z_i,z_j)big)_{i,j=1}^n$

is called the Pick matrix, and it plays a central role in various interpolation problems on various spaces.

I learned this material from Agler and McCarthy’s monograph [AM], so the following is my adaptation of that source.

(A very interesting article by John McCarthy on Pick’s theorem can be found here).

Continue reading Advanced Analysis, Notes 17: Hilbert function spaces (Pick’s interpolation theorem)