New paper: "On the matrix range of random matrices"

Malte Gerhold and I recently posted our new paper “On the matrix range of random matrices” on the arxiv, and I want to write a few words about it.

Recall that the matrix range of a $d$-tuple of operators $A = (A_1, \ldots, A_d) \in B(H)^d$ is the noncommutative set $W(A) = \cup_n W_n(A)$, where

$W_n(A) = \{ (\phi(A_1), \ldots, \phi(A_d)) : \phi : B(H) \to M_n$ is UCP $\}$.

The matrix range appeared in several recent papers of mine (for example this one), it is a complete invariant for the unital operator space generated by $A_1 \ldots, A_d$, and is within some classes is also a unitary invariant.

The idea for this paper came from my recent (last couple of years or so) flirt with numerical experiments. It has dawned on me that choosing matrices randomly from some ensembles, for example by setting

G = randn(N);

X = (G + G')/sqrt(2*N);

(this is the GOE ensemble) is a rather bad way for testing “classical” conjectures in mathematics, such as what is the best constant for some inequality. Rather, as $N$ increases, random $N \times N$ behave in a very “structured” way (as least in some sense). So we were driven to try to understand, roughly what kind of operator theoretic phenomena do we tend to observe when choosing random matrices.

The above paragraph is a confession of the origin of our motive, but at the end of the day we ask and answer honest mathematical questions with theorems and proofs. If $X^N = (X^N_1, \ldots, X^N_d)$ is a $d$-tuple of $N \times N$ matrices picked at random according to the Matlab code above, then experience with the law of large numbers, the central limit theorem, and Wigner’s semicircle law, suggests that $W(X^N)$ will “converge” to something. And by experience with free probability theory, if it converges to something, then is should be the matrix range of the free semicircular tuple. We find that this is indeed what happens.

Theorem: Let $X^N$ be as above, and let $s = (s_1, \ldots, s_d)$ be a semicircular family. Then for all $n$,

$\lim_{N \to \infty} d_H(W_n(X^N),W(s)) = 0$ almost surely

in the Hausdorff metric.

The semicircular tuple $s = (s_1, \ldots, s_d)$ is a certain $d$-tuple of operators that can be explicitly described (see our paper, for example).

We make heavy use of some fantastic results in free probability and random matrix theory, and our contribution boils down to finding the way to use existing results in order to understand what happens at the level of matrix ranges. This involves studying the continuity of matrix ranges for continuous fields of operators, in particular, we study the relationship between the convergence

(*) $\lim_{N \to \infty} \|p(X^N)\| = \|p(X)\|$

(which holds for $X^N$ as above and $X = s$ by a result of Haagerup and Torbjornsen) and

(**) $\lim_{N \to \infty} d_H(W_n(X^N),W(X)) = 0$.

To move from (*) to (**), we needed to devise a certain quantitative Effros-Winkler Hahn-banach type separation theorem for matrix convex sets.

Boris Paneah (1936-2019)

Last Thursday (10.10.2019) my beloved teacher and master’s thesis advisor Boris Paneah passed away. Boris was a great mathematician who worked most of his career on PDEs and harmonic analysis. In the last decade of his mathematical activity, he founded a new direction in the theory of functional equations, using dynamical methods. His publications can be found on his homepage. For me, he was most of all a very special teacher. In fact, he played an important role in my decision to continue to graduate studies. I want to share here an obituary that I wrote for him (it is in Hebrew).

The disc trick (and some other cute moves)

This post is about a chain of little tricks that I discovered with collaborators and used in several papers. It is just a collection of simple moves that lets you deduce the existence of a zero preserving map of a certain class between two gauge invariant spaces, given the existence of a map from that class (things will be very clear soon, I hope). These tricks were later used by some other people, who applied it in different settings.

I am writing this post as notes for my upcoming Pizza & Beer seminar talk. The section at the end of the notes contains references and links to papers where this was used.

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The complex matrix cube problem – new results from summer projects

In this post I will summarize the results obtained by my group in the “Summer Projects Week” that took place two weeks ago at the Technion. As in last time (see here for a summary of last year’s project) the title of the project I suggested was “Numerical Explorations of Open Problems from Operator Theory”. This time, I was lucky to have Malte Gerhold and Satish Pandey, my postdocs, volunteer to help me with the mentoring. The students who chose our project were Matan Gibson and Ofer Israelov, and they did fantastic work.

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The perfect Nullstellensatz just got more perfect

After giving a talk about the perfect Nullstellensatz (the commutative free Nullstellensatz) at the Technion Math department’s pizza and beer seminar, I had a revelation: I think it holds over other fields as well, not just over the complex numbers! (And in particular, contrary to what I thought before, it holds over the reals. It seems to hold over other fields as well).

To explain, I will need some notation.

Let $k$ be a field. We write $A = k[z_, \ldots, z_d]$ – the algebra of all polynomials in $d$ (commuting) variables over the field $k$