### A proof of Holder’s inequality

One of the parts of this blog that I am most proud of is my series of “Souvenirs” post, where I report about my favorite new finds in conferences. In July I went to a big conference (IWOTA 2016 in St. Louis) that I was looking forward to going to for a long time, but I did not write anything after I returned. It’s not that there was nothing to report – there was a lot, it was great conference. I was just too busy with other things.

Why am so busy? Besides being the father of seven people (most of them “kids”) and preparing for next year, I am in the last stages of writing a book, partly based on the lecture notes on “Advanced Analysis” that appeared in this blog, and on lecture notes that evolved from that. (When it will be ready I will tell you about it, you can be sure). I want to share here and now one small excerpt from it (thanks to Guy Salomon for helping me finesse it!)

Working on the final touches to the book, I decided to include a proof of Holder’s inequality in it, but I did not want to copy a proof from somewhere.  So I came up with the following proof, which I think is new (and out of curiosity I am asking you to please tell me if you have seen it before). The lazy idea of the proof is to use the fact that we already know – thanks to Cauchy-Schwarz – that the inequality holds in the $p =2$ case, and to try to show how the general case follows from that.

In other words, instead of bringing you fancy souvenirs from St. Louis, I got you this little snack from the nearby mall (really, the proof crystallized in my head when my daughter, my dog and I were sitting and waiting on a bench in the mall until other members of our family finish shopping).

Definition. Two extended real numbers $p,q \in [1, \infty]$ are said to be conjugate exponents if

$\frac{1}{p} + \frac{1}{q} = 1$.

If $p=1$ then we understand this to mean that $q = \infty$, and vice versa.

For any (finite or infinite) sequence $x_1, x_2, x_3, \ldots$, and and any $p \in [1,\infty]$, we denote

$\|x\|_p =\big(\sum |x_k|^p \big)^{1/p}$.

Theorem (Holder’s inequality): Let $p,q \in [1, \infty]$ be conjugate exponents.
Then for any two (finite or infinite) sequences $x_1, x_2, \ldots$ and $y_1, y_2, \ldots$

$\sum_k |x_k y_k| \leq \|x\|_p \|y\|_q.$

Proof. The heart of the matter is to prove the inequality for finite sequences. Pushing the result to infinite sequences does not require any clever idea, and is left to the reader (no offense).
Therefore, we need to prove that for every $x = (x_k)_{k=1}^n$ and $y = (y_k)_{k=1}^n$ in $\mathbb{C}^n$,

(HI)   $\sum |x_ky_k| \leq \big(\sum |x_k|^p \big)^{1/p} \big( \sum |y_k|^q \big)^{1/q}$.

The case $p=1$ (or $p=\infty$) is immediate. The right hand side of (HI) is continuous in $p$ when $x$ and $y$ are held fixed, so it enough to verify the inequality for a dense set of values of $p$ in $(1,\infty)$.

Define

$S = \Big\{\frac{1}{p} \in (0,1) \Big| p$ satisfies  (HI)  for all $x,y \in \mathbb{C}^n \Big\}$.

Now our task reduces to that of showing that $S$ is dense in $(0,1)$. By the Cauchy-Schwarz inequality, we know that $\frac{1}{2} \in S$. Also, the roles of $p$ and $q$ are interchangeable, so $\frac{1}{p} \in S$ if and only if $1 - \frac{1}{p} \in S$.

Set $a = \frac{q}{2p+q}$ ($a$ is chosen to be the solution to $2ap = (1-a)q$, we will use this soon). Now, if $\frac{1}{p} \in S$, we apply (HI) to the sequences $(|x_k| |y_k|^{a})_k$ and $(|y_k|^{1-a})_k$, and then we use the Cauchy-Schwarz inequality, to obtain

$\sum |x_k y_k| = \sum|x_k||y_k|^a |y_k|^{1-a}$

$\leq \Big(\sum |x_k|^p |y_k|^{ap} \Big)^{1/p}\Big(\sum |y_k|^{(1-a)q} \Big)^{1/q}$

$\leq \Big((\sum |x_k|^{2p})^{1/2} (\sum |y_k|^{2ap})^{1/2} \Big)^{1/p}\Big(\sum |y_k|^{(1-a)q} \Big)^{1/q}$

$= \Big(\sum |x_k|^{p'} \Big)^{1/p'} \Big(\sum|y_k|^{q'} \Big)^{1/q'}$

where $\frac{1}{p'} = \frac{1}{2p}$ and $\frac{1}{q'} = \frac{1}{2p} + \frac{1}{q}$. Therefore, if $s = \frac{1}{p} \in S$, then $\frac{s}{2} = \frac{1}{2p} \in S$; and if $s = \frac{1}{q} \in S$, then $\frac{s+1}{2} = \frac{1}{2}\frac{1}{q}+\frac{1}{2} = \frac{1}{q} + \frac{1}{2}\frac{1}{p}$ is also in $S$.

Since $\frac{1}{2}$ is known to be in $S$, it follows that $\frac{1}{4}$ and $\frac{3}{4}$ are also in $S$, and continuing by induction we see that for every $n \in \mathbb{N}$ and $m \in \{1,2, \ldots, 2^n-1\}$, the fraction $\frac{m}{2^n}$ is in $S$. Hence $S$ is dense in $(0,1)$, and the proof is complete.

### Dilations, inclusions of matrix convex sets, and completely positive maps

In part to help myself to prepare for my talk in the upcoming IWOTA, and in part to help myself prepare for getting back to doing research on this subject now that the semester is over, I am going to write a little exposition on my joint paper with Davidson, Dor-On and Solel, Dilations, inclusions of matrix convex sets, and completely positive maps. Here are the slides of my talk.

The research on this paper began as part of a project on the interpolation problem for unital completely positive maps*, but while thinking on the problem we were led to other problems as well. Our work was heavily influenced by works of Helton, Klep, McCullough and Schweighofer (some which I wrote about the the second section of this previous post), but goes beyond. I will try to present our work by a narrative that is somewhat different from the way the story is told in our paper. In my upcoming talk I will concentrate on one aspect that I think is most suitable for a broad audience. One of my coauthors, Adam Dor-On, will give a complimentary talk dealing with some more “operator-algebraic” aspects of our work in the Multivariable Operator Theory special session.

[*The interpolation problem for unital completely positive maps is the problem of finding conditions for the existence of a unital completely positive (UCP) map that sends a given set of operators $A_1, \ldots, A_d$ to another given set $B_1, \ldots, B_d$. See Section 3 below.]

### Thirty one years later: a counterattack on Halmos’s critique of non-standard analysis

As if to celebrate in an original way the fifty year anniversary of Bernstein and Robinson’s solution to (a generalization of) the Smith-Halmos conjecture (briefly, that if $T$ is an operator such that $p(T)$ is compact for some polynomial $p$, then $T$ has an invariant subspace), several notable mathematicians posted a interesting and very nonstandard (as they say) paper on the arxiv.

This paper briefly tells the story regarding the publication of this paper, in which Bernstein and Robinson use Robinson’s new theory of non-standard analysis (NSA) to prove the above mentioned conjecture in operator theory. This was one of the first major successes of NSA, and perhaps one would think that all of the operator theory community should have accepted the achievement with nothing but high praise. Instead, it was received somewhat coldly: Halmos went to work immediately to translate the NSA proof and published a paper proving the same result, with a proof in “standard” operator theoretic terms. (See the paper, I am leaving out the juicy parts). And then, since 1966 until 2000 (more or less), Halmos has been apparently at “war” with NSA (in the paper the word “battle” is used), and has also had criticism of logic; for example, it is implied in his book that he did not always consider logic to be a part of mathematics, worse, it seems that he has not always considered logicians to be mathematicians. (When I wrote about Halmos’s book a few months ago, I wrote that I do not agree with all the opinions expressed in the book, and I remember having the issue with logic and logicians in my mind when writing that).

In the paper that appeared on the arxiv today, the authors take revenge on Halmos. Besides a (convincing) rebuttal of Halmos’s criticisms, the seven authors hand Halmos at least seven blows, not all of them below the belt. The excellent and somewhat cruel title says it all: A non-standard analysis of a cultural icon: the case of Paul Halmos.

Besides some feeling of uneasiness in seeing a corpse being metaphorically stabbed (where have you been in the last thirty years?), the paper raises interesting issues (without wallowing too much on either one), and may serve as a lesson to all of us. There is nothing in this story special to operator theory versus model theory, or NSA, or logic. The real story here is the suspicion and snubbish-ness of mathematicians towards fields in which they do not work, and towards people working in these fields.

I see it all the time. Don’t kid me: you have also seen quite a lot of it. It is possible, I confess, that I have exercised myself a small measure of suspicion and contempt to things that I don’t understand. As the authors of the paper hint, these things are worse than wrong – they might actually hurt people.

Anyway, many times people who are ignorantly snobbish to other fields end up looking like idiots. Stop doing that, or thirty years from now a mob of experts will come and tear you to shreds.

P.S. – It seems that the question of who was the referee of the Bernstein-Robinson paper is not settled, though some suspect it was Halmos. Well, if someone could get their hands on the (anonymous!) referee report (maybe Bernstein or Robinson kept the letter?), I am quite sure that if it was Halmos, it would be clear. In other words, if Bernstein or Robinson suspected that it was him on account of the style, then I bet it was.

P.P.S. – regarding the theorem starting this discussion the quickest way to understand it is via Lomonosov’s theorem. The invariant subspace theorem proved by Bernstein and Robinson (polynomially compact operator has an invariant subspace) is now superseded by Lomonosov’s theorem (google it for a simple proof), which says that every bounded operator on a Banach space that commutes with a nonzero compact operator has a non-trivial invariant subspace.

### Summer projects in math at the Technion 2016

This year, the Faculty of Math at the Technion is continuing with its recently founded tradition of summer projects. As in last year’s week of summer projects, the Faculty of Math at the Technion is inviting advanced undergraduates from Israel and from around the world to get a little taste of research in mathematics. This is a nice opportunity, especially for someone who is considering graduate studies in math.

For a list of topics with abstracts, and for other important details (like how to apply), see this page.

### Revising and resubmitting my opinions on refereeing

With time, with age, having done already quite a few paper-refereeing jobs, I have come to change some of my opinions on refereeing.

Anonymous refereeing. I used to think that anonymous refereeing was not important. Why can’t I (as referee) just write back to the authors and discuss the weak points of the paper with them. Wouldn’t that be much better and faster? Besides, if I have a certain opinion about a paper, I should be willing to back it with my name, publicly.

Yes, I was innocent and was not yet aware of the endless ways in which some people will try to get back at you if your report includes anything but praise and/or typo corrections. But besides the usual reasons for or against anonymous refereeing, here is something I overlooked.

The really nice thing about anonymous refereeing is this: not only does it free the referee to say bad things, it also frees the referee to say good things. There was a paper I was reviewing for a good journal, and I really wanted the paper to get accepted. I thought it was very good, and that it should be accepted to this good journal. I wanted to be very clear that this paper should be accepted (sometimes, a lukewarm report is not enough to get a paper accepted), and being anonymous made it easier for me to use superlatives that I rarely feel comfortable using in front of someones face. The fact that I was anonymous, and the fact that the editor knows that I am anonymous, also makes it easier for the editors to take my praise seriously.

Therefore, the identity of referees should be kept secret, so we can all be kind to each other. (And please don’t ever ask me if it was me who refereed your paper).

Is the paper interesting? When I first heard that papers get rejected because they are “not interesting”, I was a little surprised. “Interesting” is not an objective criterion. It might be interesting to one person, and not interesting to another. Certainly the author thinks it is interesting!

Certainly? Well, I have seen some papers, unfortunately, about which I cannot say that I am certain that even the author thought that they are interesting. I have seen some papers that were written only because they could be written. Nobody ever wrote that particular proof to that particular proposition, with this set of assumptions, so this is a “new contribution”. But sometimes, a paper contains nothing which has appeared before, but does not really contain anything new. If there is nothing new, then it is boring, not interesting.

It is very hard to say what makes a good paper, and what makes a bad one. What makes good scientific research? I believe that judging the value of scientific research is not a scientific activity in itself. Deciding whether a mathematical paper is good is a job for mathematicians, but it is not a mathematical problem.

So when I evaluate a paper, I check if it is correct and new, of course, but I also cannot help but thinking whether or not it is interesting. What does interesting mean? It means interesting to me, of course! But that’s OK, because if the editor asked for my opinion, then it is my opinion that I am going to give.

Do I work for the journal. The editors of Journal A say that they want to publish only the best research articles. What does that mean? How can I compare? Let me tell you if the paper is new, correct, and interesting. What do I care that Journal A wants to remain prestigious? In fact, I never published in Journal A, and as far as I care its reputation can go to hell.

And really, to be honest, there are many factors that may affect my decision to recommend acceptance of  a paper to Journal A: 1) The authors are young researchers and this could help them in their career. 2) The paper is in my field, and I want to use the reputation of Journal A to increase the prestige of my field. 3) etc., etc., one can think of all kinds of impure reasons to be consciously biased for accepting a paper. In any case, if the paper is in my opinion a good, solid contribution, then why is it my business that the editor wants only to publish spectacular papers?

I now look at it differently. It is an honour to be approached by Journal A and be asked for their opinion. The editor is asking my professional opinion, based on my reputation. I should keep in mind that my answer, among other things, affects my reputation. I have to behave like a professional, and answer the question asked. Of course, I still don’t work for the journal, and I am free to be very enthusiastic about papers that are important in my opinion.

More on the pecking order. I have heard more than once of the following scenario: an editor of Journal B tells a referee that his journal (Journal B) is now only accepting papers that would be good enough for Journal A.

Excuse me!? If the authors thought their paper was good enough for Journal A, then they would submit it to Journal A, and not to B! And anyway, I don’t work for the journal! Clearly the journal has its goals, it wants to increase its prestige (or whatever), but I also have my own priorities, and in any case I don’t care about the prestige of Journal B. I’m already being very nice that I am willing to referee this paper for free, so don’t ask me to work for your prestige (if it is good enough for me to referee, then it’s good enough for you to publish).

Actually, the idea that Journal B aims to be at the “quality” of Journal A (whatever that means) is not so ridiculous. Journal A rejects most of the papers submitted to it, in fact it rejects some excellent papers. Where are all these papers supposed to go? So I don’t mind answering the question asked. (What I once wanted to write, but did not, is this: “Yes, I would recommend it for Journal A, and in fact this paper is too good for you, Journal B! I recommend rejecting the paper on the grounds that it is too good for this journal…”)

Submitting my review in a timely manner. I have not changed my mind about that. I always give an estimate of when I will submit my report, and I always submit on (or before) time. This means that I have to say “no” to a large fraction of referee requests (I try to referee at least about as many papers as I publish every year), otherwise I would not be able to do it in a timely manner. Naturally, I try to accept for review the papers that are more interesting.

### Hal-moss, not Hal-mush

The title of this post is a small service to Paul Halmos. I recently read his book “I Want to be a Mathematician”, subtitled “an Automathography”, where I found this:

Do all readers know that I reject ‘Hal-mush’ – some people’s notion of the “right” way to pronounce me? Please, please, say ‘Hal-moss’.

Sure, as you wish (I occasionally used to say “Hal-mosh”. No more).

Halmos was an influential mathematician who was born a hundred years ago, and died ten years ago. He worked in several areas (measure and ergodic theory, logic, operator theory) and wrote many successful books. He is considered to be a superb expositor. [His Hilbert Space Problem Book is the most refreshing, provocative and captivating book that I ever found accidentally on the library shelf (browsing with no definite goal, when I was a TA in a course on functional analysis). A Hilbert Space Problem Book is not only a beautiful and original idea, it is also executed to perfection and thus very useful.]

Perhaps I will take the opportunity of his 100 hundredth birthday (a few months from now) to write about one or some of his classic papers. But now I want to write about the book “I Want to be a Mathematician”.

The book (the automathography) is a kind of professional autobiography, omitting almost everything in personal life, and concentrating on his life as a mathematician, and that includes almost every aspect of the profession. Halmos has some interesting and definite opinions on various matters, and he believes that they should be expressed unequivocally: “I must not waffle and shilly-shally. It’s better to be wrong sometimes than to equivocate…”. The readers follow Halmos’s career, and every mathematician who crossed his way (including himself) is given a supposedly fair yet ruthless treatment. This is great book.

Halmos is happy to sort mathematicians into ranks: Gauss and Archimedes are mathematicians of the first rank, Klein and MacLane the second, Mackey, Tarski and Zygmund the third. He puts himself in the fourth rank, together with Birkhoff and Kuratowski. Immediately after discussing ranks, he introduces Fomin, a mathematician who he met in Moscow: “As a mathematician, he was perhaps of rank five”.

The final chapter of the book is called “How to be a mathematician”. I am guessing (a wild guess) that Halmos considered this as a possibility for the title of the book, but realised that it’s the wrong title. A more precise title for the book would be “How to be Halmos”. In the ruthless spirit of the author, one might also suggest: “How to be a great mathematician without really being one”.

Perhaps that’s precisely what makes the book so interesting to me. It is written by an unquestionably human mathematician. Smart, innovative, talented, idiosyncratic, hard working, ambitious – yes, but still human. An important mathematician, but not a Great One. I recommend it, it is fun to read whether or not you agree with what he has to say. I have a lot of criticism on his views, but man does he know how to write!

(Well, the book is perhaps too long and has it’s ups and downs. But one is free to skip the boring “funny” stories on the incompetent waiter in Moscow, or adventures in Uruguay).

I cannot resist objecting loudly to two pieces of advice that Halmos gives.

Halmos writes “…to stay  young, you have to change fields every five years.” Watch out (everyone except Terry, yes?): that is dangerous advice!

I personally love to branch out and work on different kinds of problems, and to learn things in different fields, but if you are interested in reaching into the deep you have to focus on some concentrated part of mathematics for a long time, for years. I have no regrets, but my experience taught me a few things that one should take into account. When you switch fields the expertise which you acquired becomes pretty much useless and you have to invent or learn new techniques from scratch. To become a reliable scholar in a new area you have to pay an expensive entrance fee by learning the literature, and your investment in the literature of the previous field goes to waste, at least in some sense. From a pragmatic point of view, it will be hard to get good letters for your promotion if you don’t stick long enough in one field to make an impact. And you may receive invitations to workshops and conferences that are no longer very relevant to you, while you are not yet recognised by the people organising workshops that you would like to go to.

It is very hard to be a true expert, a learned scholar, and to make an impact even in one field. Halmos worked on measure theory, ergodic theory, probability, statistics, operator theory, and logic. It is very very unusual, and I don’t know if Halmos is really an exception, for someone who is even very strong to make deep contributions in logic as well as in operator theory. Well, at least in this time and age it is very unusual – remember that Halmos was born 100 years ago, and mathematics has changed since the 40s and 50s quite a lot. But I think that changing fields dramatically and often was bad advice even when Halmos was active. Would his contributions to operator theory been deeper if he had not left it for several years to work on logic?

Of course, if an opportunity to branch out comes along, if your heart pulls you to a different subject, if one problem leads you naturally into a different field, then go for it! But changing fields is not an item on your checklist. Contrary to what Halmos writes, “if a student writes a thesis on the calculus of variations when he is 25, and keeps publishing papers on the calculus of variations till he is 65”, he certainly may be a first rate mathematician.

The second piece of Halmos wisdom I wish to denounce is something that appears in the chapter “How to be a mathematician”, a piece which has appeared separately and which I bumped into already many years ago, and has annoyed me even then.

Halmos writes: “[to be a mathematician] you must love mathematics more than anything else”. He goes on:

To be a mathematician you must love mathematics more than family, religion, money, comfort, pleasure, glory.

What!? More than your children? Well, Halmos did not have any children, and he probably would not have written that line if he did. But even if you don’t have children, really? Do you love mathematics more than love? More than making love? I reject this point of view altogether.

Sure, it’s not just “a job”. You shouldn’t (and couldn’t) be a mathematician if you are not thrilled by it, if it does not captivate your thoughts sometimes to the point of obsession. And you won’t succeed unless you are very devoted, unless you work with joy and work very hard. But if math is more important to you than everything else, then you are simply nuts. It can’t be more important to you more than everything else, because it’s not. In any case, there are many counter examples to the above assertion; many (all?) great mathematicians had loves, devotions, or callings, bigger than mathematics.

In fact, I believe that Halmos himself is a counter example to his claim. You can find the proof in the first and last few paragraphs of the book. These are among the most touching passages in the book, so I will just leave it at that.

*****

Apropos Halmos’s book, I take this opportunity to NOT recommend – meaning recommend not to read – Hardy’s book “A Mathematician’s Apology” (Prof. Hardy: apology not accepted!) together with Littlewood’s “A Mathematical Miscellany” (who cares?). I read Hardy’s book because a friend recommended it very highly, and I read Littlewood’s book as a possible compensation, or better: retaliation, for reading Hardy’s book. My verdict: bad books, don’t waste your time with either of these!

### Preprint update (Stable division and essential normality…)

Shibananda Biswas and I recently uploaded to the arxiv a new version of our paper “Stable division and essential normality: the non-homogeneous and quasi-homogeneous cases“. This is the paper I announced in this previous post, but we had to make some significant changes (thanks to a very good referee) so I think I have to re announce the paper.

I’ve sometimes been part of conversations where we mathematicians share with each other stories of how some paper we wrote was wrongfully (and in some cases, ridiculously) rejected; and then I’ve also been in conversations where we share stories of how we, as referees, recently had to reject some wrong (or ridiculous) paper. But I never had the occasion to take part in a conversation in which authors discuss papers they wrote that have been rightfully rejected. Well, thanks to the fact that I sometimes work on problems related to Arveson’s essential normality conjecture (which is notorious for having caused some embarrassment to betters-than-I), and also because I have become a little too arrogant and not sufficiently careful with my papers, I have recently become the author of a rightfully rejected paper. It is a good paper on a hard problem, I am not saying it is not, and it is (now (hopefully!)) correct, but it was rejected for a good reason. I think it is a story worth telling. Before I tell the story I have to say that both the referee and my collaborator were professional and great, and this whole blunder is probably my fault.

So Shibananda Biswas and I submitted this paper Stable division and essential normality: the non-homogeneous and quasi-homogeneous cases for publication. The referee sent back a report with several good comments, two of which turned out to be serious. The two serious comments concerned what appeared as Theorem 2.4 in the first version of the paper (and it appears as the corrected Theorem 2.4 in the current version, too). The first serious  issue was that in the proof of the main theorem we mixed up between $t$ and $t+1$, and this, naturally, causes trouble (well, I am simplifying. Really we mixed between two Hilbert space norms, parametrised by $t$ and $t+1$). The second issue (which did not seem to be a serious one at first) was that at some point of the proof we claimed that a particular linear operator is bounded since it is known to be bounded on a finite co-dimensional subspace; the referee asked for clarifications regarding this step.

The first issue was serious, but we managed to fix the original proof, roughly by changing $t+1$ back to $t$. There was a price to pay in that the result was slightly weaker, but not in a way that affected the rest of the paper. Happily, we also found a better proof of the result we wanted to prove in the first place, and this appears as Theorem 2.3 in the new and corrected version of the paper.

The second issue did not seem like a big deal. Actually, in the referee’s report this was just one comment among many, some of which were typos and minor things like that, so we did not really give it much attention. A linear operator is bounded on a finite co-dimensional subspace, so it is bounded on the whole space, I don’t have to explain that!

We sent the revision back, and after a while the referee replied that we took care of most things, but we still did not explain the part about the operator-being-bounded-because-it-is-bounded-on-a-finite-co-dimensional-space. The referee suggested that we either remove that part (since we already had the new proof), or we explain it. The referee added, that in either case he suggests to accept the paper.

Well, we could have just removed that part indeed and had the paper accepted, but we are not in the business of getting papers accepted for publication, we are in the business of proving theorems, and we believed that our original proof was interesting in itself since it used some interesting new techniques. We did not want to give up on that proof.

My collaborator wrote a revision with a very careful, detailed and rigorous explanation of how we get boundedness in our particular case, but I was getting angry and I made the big mistake of thinking that I am smarter than the referee. I thought to myself: this is general nonsense! It always holds. So I insisted on sending back a revision in which this step is explained by referring to a general principle that says that an operator which is bounded on a finite co-dimensional subspace is bounded.

OOPS!

That’s not quite exactly precisely true. Well, it depends what you mean by “bounded on a finite co-dimensional subspace”. If you mean that it is bounded on a closed subspace which has a finite dimensional algebraic complement then it is true, but one can think of interpretations of “finite co-dimensional” that make this is wrong: for example, consider an unbounded linear functional: it is bounded on its kernel, which is finite co-dimensional in some sense, but it is not bounded.

The referee, in their third letter, pointed this out, and at this point the editor decided that three strikes and we are out. I think that was a good call. A slap in the face and a lesson learned. I only feel bad for my collaborator, since the revision he prepared originally was OK.

Anyway, in the situation studied in our paper, the linear subspace on which the operator is bounded is a finite co-dimensional ideal in the ring of polynomials. It’s closure has zero intersection with the finite dimensional complement (the proof of this is not very hard, but is indeed non-trivial and makes use of the nature of the spaces in question), and everything is all right.  Having learned our lessons, we explain everything in detail in the current version. I hope that carefully enough.

I think that what caused us most trouble was that I did not understand what the referee did not understand. I assumed (very incorrectly, and perhaps arrogantly) that they did not understand a basic principle of functional analysis; it turned out that the referee did not understand why we are in a situation where we can apply this principle, and with hindsight this was worth explaining in more detail.

### Topic in Operator Theory 106435 (Spring 2016)

I am happy to announce that in spring semester I will be teaching a “topics” course in operator theory here at the Technion. This course is a graduate course, and is suitable for anyone who took the graduate course in functional analysis and enjoyed it. This is not exactly a course in C*-algebras, and not exactly a course in operator theory, but rather very particular blend of operator theory and operator algebras that starts from very basic material but proceeds very quickly (covering “Everything one should know”), and by the end of the course I hope some of the students will be able to consider research in this area. The official information page is available here.

I might use this blog for posting information or even notes from time to time. Here is the first piece of important information:

I will assume that students taking the course know some basic Banach algebra theory, including commutative Banach and C*-algebras. If you are interested in taking the course, make sure you know this stuff. A good source from which you can refresh your memory is Arveson’s book “A Short Course on Spectral Theory”. Start reading at the start all the way up to Section 2.3.

### Souvenirs from Bangalore 2015

Last week I attended the conference “Complex Geometry and Operator Theory” in Indian Statistical Institute, Bangalore. The conference was also an occasion to celebrate Gadadhar Misra‘s 60s birthday.

As usual for me in conferences, I played a game with myself in which my goal was to find the most interesting new thing I learned, and then follow up on it to some modest extent. Although every day of the three day conference had at least two excellent lectures that I enjoyed, I have to pick one or two things, so here goes.

#### 1. Noncommutative geometric means

The most exciting new-thing-I-learned was something that I heard not in a lecture but rather in a conversation I had with Rajendra Bhatia in one of the generously long breaks.

A very nice exposition of what I will briefly discuss below appears in this expository paper of Bhatia and Holbrook.

The notion of arithmetic mean generalizes easily to matrices. If $A,B$ are matrices, then we can define

$M_a(A,B) = \frac{A+B}{2}$.

When restricted to hermitian matrices, this mean has some expected properties of a mean. For example,

1. $M_a(A,B) = M_a(B,A)$,
2. If $A \leq B$, then $A \leq M_a(A,B) \leq B$,
3. $M_a(A,B)$ is monotone in its variables.

A natural question – which one may ask simply out of curiosity – is whether the geometric mean $(x,y) \mapsto \sqrt{xy}$ can also be generalized to pairs of positive definite matrices. One runs into problems immediately, since if $A$ and $B$ are positive definite, one cannot extract a “positive square root” from $AB$, since when $A$ and $B$ do not commute then their product $AB$ need not be a positive matrix.

It turns out that one can define a geometric mean as follows. For two positive definite matrices $A$ and $B$, define

(*) $M_g(A,B) = A^{1/2} \sqrt{A^{-1/2} B A^{-1/2}} A^{1/2}$ .

Note that when $A$ and $B$ commute (equivalently, when they are scalars) then $M_g(A,B)$ reduces to $\sqrt{AB}$, so this is indeed a generalisation of the geometric mean. Not less importantly, it has all the nice properties of a mean, in particular properties 1-3 above (it is not evident that it is symmetric (the first condition), but assuming that the other two properties follow readily).

Now suppose that one needs to consider the mean of more than two – say, three – matrices. The arithmetic mean generalises painlessly:

$M_a(A,B,C) = \frac{A + B + C}{3}$.

As for the geometric mean, there has not been found an appropriate algebraic expression that generalises equation (*) above. About a decade ago, Bhatia, Holbrook and (separately) Moakher, found a geometric way to define the geometric mean of any number of positive definite matrices.

They key is that they view the set $\mathbb{P}_n$ of positive definite $n \times n$ matrices as a Riemannian manifold, where the length of a curve $\gamma : [0,1] \rightarrow \mathbb{P}_n$ is given by

$L(\gamma) = \int_0^1 \|\gamma(t)^{-1/2} \gamma'(t) \gamma(t)^{-1/2}\|_2 dt$,

where $\|\cdot\|_2$ denotes the Hilbert-Schmidt norm $\|A\|_2 = trace(A^*A)$. The length of the geodesic (i.e., curve of minimal length) connecting two matrices $A, B \in \mathbb{P}_n$ then defines a distance function on $\mathbb{P}_n$, $\delta(A,B)$.

Now, the connection to the geometric mean is that $M_g(A,B)$ turns out to be equal to the midpoint of the geodesic connecting $A$ and $B$! That’s neat, but more importantly, this gives an insight how to define the geometric mean of three (or more) positive definite matrices: simply define $M_g(A,B,C)$ to be the unique point $X_0$ in the manifold $\mathbb{P}_n$ which minimises the quantity

$\delta(A,X)^2 + \delta(B,X)^2 + \delta(C,X)^2$.

This “geometric” definition of the geometric mean of positive semidefinite matrices turns out to have all the nice properties that a mean should have (the monotonicity was an open problem, but was resolved a few years ago by Lawson and Lim).

This is a really nice mathematical story, but I was especially happy to hear that these noncommutative geometric means have found highly nontrivial (and important!) applications in various areas of engineering.

In various engineering applications, one makes a measurement such that the result of this measurement is some matrix. Since measurements are noisy, a first approximation for obtaining a clean estimate of the true value of the measured matrix, is to repeat the measurement and take the average, or mean of the measurements. In many applications the most successful (in practice) mean turned out to be the geometric mean as described above. Although the problem of generalising the geometric mean to pairs of matrices and then to tuples of matrices was pursued by Bhatia and his colleagues mostly out of mathematical curiosity, it turned out to be very useful in practice.

#### 2. The Riemann hypothesis and a Schauder basis for $\ell^2$.

I also have to mention Bhaskar Bagchi’s talk, which stimulated me to go and read his paper “On Nyman, Beurling and Baez-Duarte’s Hilbert space reformulation of the Riemann hypothesis“. The main result (which is essentially an elegant reformulation of a quite old result of Nyman and Beurling, see this old note of Beurling)  is as follows. Let $H$ be the weighted $\ell^2$ space given by all sequence $(x_n)_{n=1}^\infty$ such that

$\sum_n \frac{|x_n|^2}{n^2} < \infty$.

In $H$ consider the sequence of vectors:

$\gamma_2 = (1/2, 0, 1/2, 0, 1/2, 0,\ldots)$

$\gamma_3= (1/3, 2/3, 0, 1/3, 2/3, 0, 1/3, 2/3, 0,\ldots)$

$\gamma_4 = (1/4, 2/4, 3/4, 0, 1/4, 2/4, 3/4, 0, \ldots)$

$\gamma_5 = (1/5, 2/5, 3/5, 4/5, 0, 1/5, \ldots)$,

etc. Then Bagchi’s main result is

Theorem: The Riemann Hypthesis is true if and only if the sequence $\{\gamma_2, \gamma_3, \ldots \}$ is total in $H$

This is interesting, though such results can always be interpreted simply as a claim that the necessary and sufficient condition is now provenly hard. Clearly, nobody expects this to open up a fruitful path by which to approach the Riemann hypothesis, but it gives a nice perspective, as Bagchi writes in his paper:

[The theorem] reveals the Riemann hypothesis as a version of the central theme of harmonic analysis: that more or less arbitrary sequences (subject to mild growth restrictions) can be arbitrarily well approximated by superpositions of a class of simple periodic sequences (in this instance, the sequences $\gamma_k$).

### Something sweet for the new year

Tim Gowers recently announced the start of a new journal, “Discrete Analysis”. The sweet thing about this journal is that it is an arxiv overlay journal, meaning that the journal will act like most other elctronic journals with the difference that all it does in the end (after standard peer review and editorial decisions) is put up a link on its website to a certain version of the preprint on the arxiv. The costs are so low, that neither readers nor authors are supposed to pay. In the beginning, Cambridge University will cover the costs of this particular journal, and there are hopes that funding will be found later (of course, arxiv has to be funded as well, but this does not seem to incur additional costs on arxiv). The journal uses a platform called Scholastica (which does charge something, but relatively low – like \$10 per paper) so they did not have to set up their webpage and deal with that kind of stuff.

The idea has been around for several years and there are several other platforms (some of which do not charge anything since they are publicly funded) for carrying journals like this: Episciences, Open Journals. It seems like analysis, and operator theory in particular, are a little behind in these initiatives (correct me if I am wrong). But I am not worried, this is a matter of time.

The news of the baby journal made me especially happy since leaders like Gowers and Tao have been previously involved with the creation of the bad-idea-author-pay-journals Forum of Mathematics (Pi and Sigma), and it is great that their stature is also harnessed for a decent journal (which also happens to have a a nice and reasonable name).