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

#### by Orr Shalit

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 and , and this, naturally, causes trouble (well, I am simplifying. Really we mixed between two Hilbert space norms, parametrised by and ). 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 back to . 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.

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

This is a genuinely great post, Orr – and the one that should be read by everyone in the field. I write it as an (occasionally) careless author, (occasionally) stubborn referee and (occasionally) inefficient editor.

Thank you.

An interesting and honest experience!

I have several things to say regarding this post, including, in particular, regarding acceptance and rejection of papers, but I prefer to say many of them (if and) when we meet on Sunday at the well-known restaurant in the usual hour. Here I will say two things:

1. I also have papers which have been rightfully rejected, definitely. I do, however, have many others (mathematical and non-mathematical) for which it is possible to prove that this is not the case, and you have already heard a few examples.

2. Regarding the boundedness of a linear operator which is known to be bounded on a finite co-dimensional closed vector subspace: well, this claim can be generalized as the following simple proposition (whose proof is given for the sake of completeness) shows:

PROPOSITION: Suppose that is a normed space which can be represented as a (not necessarily direct) sum of two (not necessarily closed) vector subspaces and . Assume further that there exist (not necessarily linear) continuous mappings and whose sum is the identity operator . Then any additive mapping from to a normed space which is continuous on both and is also continuous on . In particular, if is a Banach space which can be represented as the direct sum of two closed vector subspaces and and if T is a linear operator which is continuous on both and , then T is bounded on .

The claim mentioned in the post follows from this particular case because the restriction of the operator to the finite dimensional complement is a linear operator acting from a finite dimensional normed space to a normed space, as is well-known, such a mapping is always continuous.

PROOF: We first prove the general statement. Let be an arbitrary point in and assume that for some sequence of elements . We want to show that . Because of the assumption and the fact that is additive we can write . Since we assume that and are continuous, we have and . Therefore, using the assumptions that is continuous on both and , that and for each , and the additivity of , we conclude from the previous lines that as required.

As for the particular case mentioned at the end of the formulation of the theorem, it follows from the general statement because if a Banach space is the direct sum of two closed vector subsapces and , then for each in the space there exist unique points and such that , and the mappings defined for all by and are linear and continuous from a well-known consequence of the open mapping theorem (which can be found, e.g., on page 37 of the book “H. Brezis. Functional analysis, Sobolev spaces and partial differential equations, Springer, NY, 2011”).

Thanks for the remark.

I have a couple of comments:

1) I wish to stress that usually the real work comes in checking that the conditions of the basic general fact apply. For example, in our paper what we did not explain was that we are in a situation to apply such a principle: the map has to be given, one needs the existence of continuous projections.

2) An aside: is that really a generalization? I claim that if it’s the exact same proof as the linear case, then it is not to be called a generalization. I call it: “the same principle”. In any case, it would be interesting to see a nontrivial example where this general version can be applied (and the spaces are not closed, the projections not linear, and the map T not linear).

1. Thanks for the comment.

2. The proposition holds with the same proof also when and are any nonempty subsets of latex X$ and not necessarily vector subspaces. So far I don’t have any example beyond the known one (direct sum of closed linear subspaces and linear projections).

3. One has freedom to use which words one wishes to use when describing various phenomena, but I am not sure that when doing so, one takes into account everything possible. More specifically, I am not sure that everything has been taken into account regarding the issue of “the same principle” issue. For instance, if one has a proof that a real continuous function defined on a closed and bounded interval attains a maximum there, and then one observes that the same theorem with the same proof holds for a continuous real function defined on closed and bounded subsets of a finite dimensional Euclidean space, can we say that this is the same theorem? And what if one later observes that the same holds in any compact metric space? It is not at all obvious to observe this extension even if in retrospective the same proof holds.

As a second example, consider the concept of zone diagrams and double zone diagrams. These are rather exotic geometric objects. Historically, it was known that zone diagrams of finitely many point sites exist and unique in the Euclidean plane and nothing beyond that was known. Then somewhere it has been established that a zone diagram of two point sites and a double zone diagram of finitely many sites exist in finite dimensional Euclidean space. Gradually, it has been observed that the same proof holds for zone diagrams of two arbitrary sites (site=subset of the space) and double zone diagrams of any number (possibly an arbitrary infinite cardinal) of sites exist (for both zone diagrams and double zone diagrams) in any metric space, and then the proof has been extended to existence in what is called an “m-space” (this is a pair of a non-empty set and a function satisfying the condition that for all $x,y\in X$.

The existence of such exotic geometric objects is not clear even in the Euclidean plane (with point sites the setting in which zone diagrams have been discussed before the above mentioned result has appeared; in this simple setting one can have some intuition regarding to what expect), so it is definitely not obvious to think that they exist (with arbitrary sites) in more general setting, and even if one may somehow manage to establish the result in a metric space setting, the attempt to go to a more general setting (including a one in which the distance function may be negative, not symmetric, and may not satisfy the triangle inequality) is not at all obvious (besides, the definition of an m-space and of a double zone diagram didn’t exist before the corresponding paper has appeared). So yes, one can call this result “the same principle” as the Euclidean space with point sites case, but, in my opinion, when doing so one misses several essential issues.

More details about the above mentioned results can be found in the following paper:

“D. Reem and S. Reich, Zone and double zone diagrams in abstract spaces, Colloquium Mathematicum, 115 (2009), 129-145, arXiv:0708.2668 [math.MG], 2007 (current version: [v2] Mon, 25 Jul 2011)

http://arxiv.org/abs/0708.2668