Noncommutative Analysis

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.


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).

One of the most outrageous open problems in operator/matrix theory is solved!

I want to report on a very exciting development in operator/matrix theory: the von Neumann inequality for 3 \times 3 matrices has been shown to hold true. I learned this from a recent paper (with the irresistible title) “The von Neumann inequality for 3 \times 3 matrices“, posted on the arxiv by Greg Knese. In this paper, Knese explains how the solution of this outstanding open problem follows from results in a paper by Lukasz Kosinski, “The three point Nevanlinna-Pick problem in the polydisc” that appeared on the arxiv about a half a year ago. Beautifully, and not surprisingly, the solution of this operator/matrix theoretic problem follows from deep new facts in complex function theory in several variables.

To recall the problem, let us denote \|A\| the operator norm of a matrix A, and for every polynomial p in d variables we denote by \|p\|_\infty the supremum norm

\|p\|_\infty = \sup_{|z_i|\leq 1} |p(z_1, \ldots, z_d)|.

A matrix A is said to be contractive if \|A\| \leq 1.

We say that d commuting contractions A_1, \ldots, A_d satisfy von Neumann’s inequality if 

(*)  \|p(A_1,\ldots, A_d)\| \leq \|p\|_\infty.

It was known since the 1960s that (*) holds when d \leq 2. Moreover, it was known that for d \geq 3, there are counter examples, consisting of d contractive 4 \times 4 matrices that do not satisfy von Neumann’s inequality. On the other hand, it was known that (*) holds for any d if the matrices A_1, \ldots, A_d are of size 2 \times 2. Thus, the only missing piece of information was whether or not von Neumann’s inequality holds or not for three or more contractive 3 \times 3 matrices. To stress the point: it was not known whether or not von Neumann’s inequality holds for three three-by-three matrices. The problem in this form has been open for 15 years  – but the problem is much older: in 1974 Kaiser and Varopoulos came up with a 5 \times 5 counter-example, and since then both the 3 \times 3  and the 4 \times 4 cases were open until Holbrook in 2001 found a 4 \times 4 counter example. You have to agree that this is outrageous, perhaps even ridiculous, I mean, three 3 \times 3 matrices, come on!

In Knese’s paper this story and the positive solution to the problem is explained very clearly and succinctly, and is recommended reading for any operator theorist. One has to take on faith the paper of Kosinski which, as Knese stresses, is where the major new technical advance has been made (though one should not over-stress this fact, because tying things together, the way Knese has done, requires a deep understanding of this problem and of the various ingredients). To understand Kosinki’s paper would require a greater investment of time, but it appears that the paper has already been accepted for publication, so I am quite confident and happy to see this problem go down.

Babies welcome in my class!

Re this news item (or here, in English):

This is so wrong.

Mothers to very young babies must be given the choice to bring their babies with them to their workplace, or to the college where they study. Obviously, if the baby starts screaming, they should step outside until the baby is quiet again.

What other possibilities are there? Not study? That’s one solution (I know a lot of very smart moms who chose that). Wait with the babies until after the career goals are met (say, age 38)? Another solution. Put a 6 week old baby in a nursery? A solution. I don’t think anybody can honestly say that there is a perfect solution for mothers. Taking a baby to work or university is also not a perfect solution, but in my opinion this solution should be accepted (not to say, encouraged) by society as a legitimate one.

Normally, a baby (especially a young, nursing baby that is held) can be very quiet for a rather long time; babies are much better behaved than the average student. Don’t worry: no one wants to hush a screaming baby and ruin everybody’s day while not being able to understand anything themselves.

If anybody feels that an occasional gurgle or murmur (or a completely silent breastfeeding mom) is disruptive to learning, perhaps they should carefuly check if that is really what is bothering them. I never heard of anyone with an annoying cough, or someone who wears short sleeves or shorter pants, or has an obnoxious attitude, or somebody who asks the instructor not to use Greek letters, etc., being asked not to enter classes because it is disruptive. Universities are about people, right? It’s nice that there’s all kinds of people of various kinds and sizes, enjoy it!


Spaces of Dirichlet series with the complete Pick property (or: the Drury-Arveson space in a new disguise)

John McCarthy and I have recently uploaded a new version of our paper “Spaces of Dirichlet series with the complete Pick property” to the arxiv. I would like to advertise the central discovery of this paper here.

Recall that the Drury-Arveson space H^2_d is the reproducing kernel Hilbert space on the open unit ball of a d dimensional Hilbert space, with reproducing kernel

k(z,w) = \frac{1}{1 - \langle z, w \rangle}.

It has the remarkable universal property that every Hilbert function space with the complete Pick property is naturally isomorphic to the restriction of H^2_\infty to a subset of the unit ball (see Theorem 6 and its corollary in this post), and consequently, every complete Pick algebra is a quotient of the multiplier algebra \mathcal{M}_\infty = Mult(H^2_\infty). To the best of my knowledge, no other Hilbert function spaces with such a universal property have been studied.

John and I discovered another reproducing kernel Hilbert space that turns out to be “the same” as the Drury-Arveson space H^2_\infty. Since the space H^2_\infty as been so well studied, it interesting to discover a new incarnation. The really interesting part is that the space we discovered is a space of analytic functions on a half plane (that is, a space of functions in one complex variable), rather than a space of analytic functions in infinitely many variables on the unit ball of a Hilbert space.

To be precise, the spaces we consider are spaces of Dirichlet series \mathcal{H}, of the form

\mathcal{H} = \{f(s) = \sum_{n=1}^\infty \gamma_n n^{-s} : \sum |\gamma_n|^2 a_n^{-1} < \infty \}.

(Here a_n is a sequence of positive numbers). These are Hilbert function spaces on some half plane that have a kernel of the form k(s,u) = \sum a_n n^{-s-\bar u}.

We first answer the question which of these spaces \mathcal{H} have the complete Pick property. This problem has a simple solution (which has been anticipated by similar results on spaces on the disc): if we denote by g(s) = \sum a_n n^{-s} the “generating function” of the space, and if we write

\frac{1}{g(s)} = \sum c_n n^{-s},

then \mathcal{H} is a complete Pick space if and only if c_n \leq 0 for all n \geq 2.

After we know to tell when these spaces are complete Pick, it is natural to ask which complete Pick spaces arise like this? We do not give a complete answer, but our surprising discovery is that things can easily be cooked up so to obtain the Drury-Arveson space H^2_d, where d can be any cardinal number in \{1,2,\ldots, \infty\}. For example, \mathcal{H} turns out to be “the same” as H^2_\infty if the kernel k is given by

k(s,u) = \frac{P(2)}{P(2) - P(2+s+\bar u)},

where P(s) = \sum_{p} p^{-s} is the prime zeta function (the sum is taken over all primes p).

 Now, I have been a little vague about what it means that \mathcal{H} is “the same” as H^2_\infty. In fact, this is a subtle question, and we devote a part of our paper what it means for two Hilbert function spaces to be the same — something that has puzzled us for a while.

What does this appearance of Drury-Arveson space as a space of Dirichlet series mean? Can we use this connection to learn something new on multivariable operator theory, or on Dirichlet series? How did the prime zeta function smuggle itself into this discussion? This requires further thought.

Stable division and essential normality: the non-homogeneous and quasi homogeneous cases

Update (January 29, 2016): paper revised, see this post


Several months ago Shibananda Biswas (henceforth: Shibu) and I posted to the arxiv our paper “Stable division and essential normality: the non-homogeneous and quasi homogeneous cases“. I was a little too busy to write about it at the time, but now that it is summer it seems like a good time to do it, since I am too busy, and I need a break from work. Nothing like going back and thinking about papers you have already written when you are overwhelmed by your current project.

Anyway, the main problem the paper I wrote with Shibu deals with, is the essential normality of submodules of various Hilbert modules (closely related to the Drury-Arveson module that I wrote about in the past: one, two, three, or if you are really asking for trouble, look at this survey). This paper is highly technical, and I want to try to explain it in a non-technical fashion. Read the rest of this entry »

A corrigendum

Matt Kennedy and I have recently written a corrigendum to our paper “Essential normality, essential norms and hyperrigidity“. Here is a link to the corrigendum. Below I briefly explain the gap that this corrigendum fills.

A corrigendum is correction to an already published paper. It is clear why such a mechanism exists: we want the papers we read to represent true facts, so false claims, as well as invalid proofs or subtle gaps should be pointed out to the community. Now, many many papers (I don’t want to say “most”) have some kind of mistake in them, but not every mistake deserves a corrigendum – for example there are mistakes that the reader will easily spot and fix, or some where the reader may not spot the mistake, but the fix is simple enough.

There are no rules as to what kind of errors require a corrigendum. This depends, among other things, on the authors. Some mistakes are corrected by other papers. I believe that very quickly some sort of mechanism – say google scholar, or mathscinet – will be able to tell if the paper you are looking up is referenced by another paper pointing out a gap, so such a correction-in-another-paper may sometimes serve as legitimate replacement for a corrigendum, when the issue is a gap or minor mistake.

There is also a question of why publish a corrigendum at all, instead of updating the version of the paper on the arxiv (and this is exactly what the moderators of the arxiv told us at first when we tried to upload our corrigendum there. In the end we convinced them that the corrigendum can stand by itself). I think that once a paper is published, it could be confusing to have a version more advanced than the published version; it becomes very clumsy to cite papers like that.

The paper I am writing about (see this post to see what its about) had a very annoying gap: we justified a certain step by citing a particular proposition from a monograph. The annoying part is that the proposition we cite does not exactly deal with the situation we deal with in the paper, but our idea was that the same proof works in our situation. We did not want to spell out the details because we considered that to be very easy, and in any case it was not a new argument. Unfortunately, the same proof does work when working with homogeneous ideals (which was what first versions of the paper treated) but in fact it is not clear if they work for non-homogeneous ideals. The reason why this gap is so annoying, is that it leads the reader to waste time in a wild goose chase: first the reader goes and finds the monograph we cite, looks up the result (has to read also a few extra pages to see he understands the setting and notation in the monograph), realises this is is not the same situation, then tries to adapt the method but fails. A waste of time!

Another problem that we had in our paper is that one requires our ideals to be “sufficiently non-trivial”. If this were the only problem we would perhaps not bother writing a corrigendum just to introduce a non-triviality assumption, since any serious reader will see that we require this.

If I try to take a lesson from this, besides a general “be careful”, it is that it is dangerous to change the scope of the paper (for us – moving form homogeneous to non-homogeous ideals) in late stages of the preparation of the paper. Indeed we checked that all the arguments work for the non-homogneous case, but we missed the fact that an omitted argument did not work.

Our new corrigendum is detailed and explains the mathematical problem and its solutions well, anyone seriously interested in our paper should look at it. The bottom line is this as follows.

Our paper has two main results regarding quotients of the Drury-Arveson module by a polynomial ideal. The first is that the essential norm in the non selfadjoint algebra associated to a the quotient module, as well as the C*-envelope, are as the Arveson conjecture predicts (Section 3 in the paper) . The second is that essential normality is equivalent to hyperrigidity (Section 4 in the paper).

Under the assumption that all our ideals are sufficiently non-trivial (and some other standing assumptions stated in the paper), the situation is as follows.

The first result holds true as stated.

For the second result, we have that hyperrigidity implies essential normality (as we stated), but the implication “essential normality implies hyperrigidity” is obtained for homogeneous ideals only.



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