### A sneaky proof of the maximum modulus principle

#### by Orr Shalit

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).

nice proof do you know any other operator theoretic proofs of function theoretic problems..the corona problem is…any other….?

Thanks!

Yes, there are several function theoretic theorems which have operator theoretic proofs. That is one of the reasons why I love this field of operator theory and complex variables. Not in all of fields where operator theory meets another field does operator theory contributes back to the other field.

The corona theorem is one. Another is Pick’s interpolation theorem: see the fourth section of lecture notes #17 under the Advanced Analysis tab above for more details. One can also prove the Schwartz Lemma with von Neumann’s inequality.

There is also Loewner’s theorem: see Donoghue’s book (“Monotone matrix functions and analytic continuation”).

[…] analysis! This paper also appeared in the American Mathematical Monthly and the author wrote a blog post about it as […]

Dear Orr, nice work.

How long did you have to wait to receive the referee’s (from the AMM) report?

Thanks. Three months.