Operator Theory: Advances and Applications, Vol. 259, 533–559 c 2017 Springer International Publishing Commutator Estimates Comprising the Frobenius Norm – Looking Back and Forth Zhiqin Lu and David Wenzel A tribute to Albrecht B¨ottcher on his 60th birthday √ Abstract. The inequality XY −YX F ≤ 2 X F Y F has some history to date. The growth of the task will be highlighted, supplemented by a look at future developments. On this way, we meet different forms and give an insight into various consequences of it. The collection of results will be enriched by introductive explanations. We also cross other fields that are important for theory and applications, and even uncover less known relationships. Mathematics Subject Classification (2010). Primary 15A45; Secondary 15-02. Keywords. Commutator, Frobenius norm, BW and DDVV conjectures. 1. Revelations We want to take the opportunity of Albrecht B¨ottcher’s sixtieth birthday to look at one particular topic that started with a vague idea about a dozen years ago. First, we had a conjecture, this conjecture was proven after quite a while, and several follow-ups extended the problem thereafter. And who, if not two of the guys responsible for developing completely dif- ferent proofs, could give you some insight into this story and an overview of the achievements. There we are: Albrecht’s long-time research associate David Wen- zel, who went this path together with him; and Zhiqin Lu, coming from geometry and opening a (to most of us) new perspective. We thank Koenraad Audenaert for some entry points into physics that unveil links to a connatural field. At first, let’s go back to the beginning. Assume two square matrices X, Y are given. We all know that, occasionally, XY and YX do not coincide. But how different can they be? For measuring the distance between them, we can take a look onto XY − YXF ,where·F is the easily computable Frobenius norm. The author Z. Lu is partially supported by an NSF grant DMS-1510232. [email protected] 534 Z. Lu and D. Wenzel The object inside is the famous commutator1. What’s the deal? It can’t be hard! Just using the triangle inequality and the sub-multiplicativity, one clearly has XY − YXF ≤ 2XF Y F . (1.1) You’re right. But, the problem is that we were unable to find only one exam- ple in which actually equality holds.2 The best (meaning “biggest” in our case) we got even from extended experiments with a computer – operating√ systematically or at random, no matter the matrix size taken – was a factor 2 instead of the trivially obtained 2. A claim was born. The one that it’s all about. Theorem 1.1 (B¨ottcher–Wenzel conjecture). Suppose n ∈ N is arbitrary. Let X and Y be two n × n matrices. Then, √ [X, Y ]F ≤ 2XF Y F . (1.2) Well, you probably want to throw in the operator norm into the discussion. As many investigations are done in this one, it seems like a fixed option. However, 01 0 −1 20 the example 10 , 10 = 0 −2 shows that we are stuck with the trivial estimate (1.1). Here, the constant can not be improved. The Frobenius norm is more selective. The commutator is someway strange. As a map turning two matrices into a third one, it is not injective. The kernel is non-zero. In particular, every matrix commutes with itself. For this reason, [X, Y ]F = [X, Y + αX]F .Inconse- √quence, we can replace the right-hand side of (1.2) by virtually any of the terms − 2 2 X F Y +αX F .Sinceα is arbitrary, choosing α = X, Y / X F is possible. 3 This is indeed a minimizer of Y +αXF and would further reduce the (squared ) upper bound (if it really should be valid) by an inner product4: 2 ≤ 2 2 − | |2 [X, Y ] F 2 X F Y F 2 X, Y . (1.3) Why should (1.2), or equivalently (1.3), be true? Here is a demonstration for n =2. ab ef Proof. Put X = cd , Y = gh . An elementary calculation delivers 2 2 − | |2 − − 2 2 X F Y F 2 X, Y XY YX F (1.4) =2|ah − de|2 + |(a + d)f − (e + h)b|2 + |(a + d)g − (e + h)c|2 which clearly is a sum of non-negative terms. 1Duetoalgebra,theLiebracketnotation[X, Y ] is a common abbreviation. 2. except for the case involving zero matrices, which is pretty lame, for obvious reasons. 3 2 1/2 This is always a good idea because of the definition AF := j,k |ajk| . simplifying transformations and avoiding an unnecessary flood of root symbols. 4 An advantage of the Frobenius over the operator norm: it is induced by the scalar product ∗ A, B := Tr(B A), where Tr C := i cii denotes the trace. [email protected] Commutator Estimates 535 So easy the last lines emerged, it must be clarified that the same idea refuses to work for matrices of size 3 × 3 and beyond. It’s not that we are unable to show anything related; an analogous statement simply is not true. Also the original attempt from [6, Theorem 4.2] cannot be transferred into higher dimensions.5 Of course, there was something that made us believe into the validity of (1.2). During the initial hunt for “utterly non-commuting” matrix pairs, we made over- 2 2 2 view plots for the ratio [X, Y ] F / X F Y F and generated pictures like these: √ Undoubtedly, the values for n = 2 may reach the constant 2 (squared, of course), but won’t go further. Yet, it is another point that catches the eye. Apparently, random size 3 matrices have difficulties in producing big commutators. The clustering effect at very small values becomes even stronger when n increases. That situation left us torn between astonishment and annoyance. On the one side, the seemingly whole-range-filling 2 × 2 matrix case (the one where actually something happens) was tackled completely. But on the other, although large sized matrices evidently are far away from making any hassles, this wasn’t usable for obtaining a general proof. And even though n = 2 later turned out to be the most interesting case, there should remain lots of stuff to do! 2. Widening the scope At the start, we had a conjecture about a general norm bound, and could prove it only in some special cases (for size 2, or if one of the matrices is normal). Well, additionally, we were able to show that (1.1) must be too weak. Mastering deter- minants√ as he usually does, Albrecht could validate the estimate with 2 replaced by 3 . that’s half way on what we wished for, at least. 5The proof relies on the matrices’ trace being spread over only two entries. [email protected] 536 Z. Lu and D. Wenzel Of course, there were also the overwhelming observations that commutators typically avoid large values. So, we wanted to give this aspect a foundation. For this, we could revert to a previous work with Grudsky [5] on a bound for norms of random vectors under a linear map. Luckily, when interested in the Frobenius norm, we can look at a matrix as if it is a vector. It was shown in [6] that, taking 2 2 2 →∞ the expected value, the ratio [X, Y ] F / X F Y F tends to zero as n , and the convergence is linear. In conclusion, the norm of the commutator is small compared to the norms of the involved matrices: − 2 1 · 2 2 E XY YX F = O n E X F Y F , (2.1) under very weak assumptions on the underlying distribution.6 The efforts afterwards concentrated on the minimal bound. It stayed open for quite a while. Then, simultaneously, three really√ distinct proofs for (1.2) saw the light of the day ([7],[26],[34]). So, the constant 2 in the inequality is now shown, and it is known to be best-possible.7 Sure, with that, only the first step on an even longer road up to the present was done. 2.1. Illuminating the representation In 1900, Hilbert presented the famous list with 23 problems that should heavily in- fluence the following century of mathematics. About the half of them is completely solved by now, and still a handful of them is way too far from being understood. A particular problem, the 17th, prepares the ground for this part. It reads as follows. Consider a multivariate polynomial or rational function. If it takes only nonnegative values over R, can it be represented as a sum of squares of rational functions? It was confirmed “only” three decades later.8 In the formulation, Hilbert already has taken into account that there definitely exist nonnegative polynomials that are not a SOS9 of other polynomials. So, finding a polynomial SOS is always on the agenda whenever a polynomial in n variables is nonnegative. The commutator inequality (1.2) is equivalent to 2 2 − − 2 ≥ 2 X F Y F XY YX F 0. On the left-hand side, there obviously is a quartic polynomial. And we claimed that it is nonnegative. The investigated form was further reduced a little more to (1.4). And in the proof for n = 2, we indeed obtained a sum of polynomial squares (actually of quartics, again). It is even the complex version. Naturally, this gave hope also for larger matrices. L´aszl´o[24] proved that, as remarked, the reduced form is not SOS.