The complex matrix cube problem (in “Summer Projects in Mathematics at the Technion”)

by Orr Shalit

Next week I will participate as a mentor in the Technion’s Summer Projects in Mathematics. The project I offered is called “Numerical explorations of open problems from operator theory”, and it suggests three open problems in operator theory where theoretical progress seems to be stuck, and for which I believe that some computer experiments can help us get a feeling of what is going on. I also hope that thinking seriously about designing experiments can help us to understand some general facets of the theory.

I have been in contact with the students in the last few weeks and we decided to concentrate on “the matrix cube problem”. On Sunday, when the week begins, I will need to present the background to the project to all participants of this week, and I have seven minutes (!!) for this. As everybody knows, the shorter the presentation, the harder the task is, and the more preparation and thought it requires. So I will take use this blog to practice my little talk.

Introduction to the matrix cube problem

This project is in the theory of operator spaces. My purpose is to give you some kind of flavour of what the theory is about, and what we will do this week to contribute to our understanding of this theory.

Let H be an n-dimensional Hilbert space (this just means: an n-dimensional inner product space over the complex numbers). Recall that H is also a normed space with norm \|h\| = \sqrt{\langle h, h \rangle}. A basic fact is that every such H is isometrically isomorphic to the space \mathbb{C}^n equipped with with standard inner product

\langle u, v \rangle = \sum_{i=1}^n u_i \overline{v_i},

which induces the Euclidean norm \|u\| = \sqrt{\sum |u_i|^2}. This means that there exists a linear isomorphism T : \mathbb{C}^n \to H that preserves the inner product, and in particular, the norm : \|Tu\| = \|u\|.

Take two linearly independent vectors u,v \in H, and construct the subspace G = \textrm{span}\{u,v\} \subseteq H. Fact: no matter how we choose u and v, G is always a 2-dimensional Hilbert space, i.e., G is isometrically isomorphic to \mathbb{C}^2 with the Euclidean norm.

Now let B(H) be the space of linear operators on H. This space is also normed space, when we give it the norm

\|A\| = \sup_{\|h\|=1}\|Ah\|.

Using the isometric isomorphism H \cong \mathbb{C}^n, we will identify B(H) with the space of n \times n matrices.

Take two linearly independent operators A, B \in B(H), and construct the subspace V = \textrm{span}\{A,B\} \subseteq B(H). As a linear space, V is isomorphic to \mathbb{C}^2. However, as a normed space V might be any one of an uncountable family of two dimensional normed spaces. For example, V can be isometrically isomorphic to \ell^2_2 = \mathbb{C}^2 or to \ell^\infty_2. On the other hand, if we are assuming that H is finite dimensional, then V is cannot isometrically isomorphic to \ell^1_2! (If we allow for infinite dimensional H, then we can get any two dimensional normed space as the span of two operators).

Understanding the normed space V= \textrm{span}\{A,B\} \subseteq B(H) boils down to computing the norm:

\|\alpha A + \beta B\|

for every \alpha, \beta \in \mathbb{C}.

It is remarkable how such a simple-minded problem is actually very difficult. What mathematicians can do in difficult situations is try to do one of the following:

  1. Experiment with examples. I cannot overstate how much it is important for the health of one’s research that examples be sought and examined. Since calculations regarding the norm of matrices of moderate size require incredibly tedious calculations, it becomes at some point obvious that we should recruit the computer to help us explore what is going on.
  2. Solve the problem for an interesting special case. For example, suppose that are operators A and B are normal and commuting. Then we know that there exists an orthonormal basis in which A = \textrm{diag}(a_1, \ldots, a_d) and B = \textrm{diag}(b_1, \ldots, b_n). Then calculation of any polynomial in A,B is easy, and in particular \|\alpha A + \beta B \| = \max\{ |\alpha a_i + \beta b_i| : i=1, \ldots, n\}.
  3. Reduce the problem to a special case. For example, the easiest case is when A and B are scalars, i.e., n=1. The case of two commuting normal operators reduces to the case of scalars ones, because (A,B) in this case decomposes as the direct sum (a_1, b_1) \oplus \cdots \oplus (a_n,b_n).

We are led to the question: can we learn something about the case of general operators by using the fact that the problem is solved for commuting normal ones?

Now, a general pair of operators cannot be decomposed into some kind of “sum” of normal commuting pairs. However, we do have the following theorem.

Theorem. There exists a constant C such that for any two n \times n matrices A,B, identified with two operators on H, there exists two commuting normal operators M,N such that

(*) M = \begin{pmatrix} A & * \\ * & * \end{pmatrix}       N = \begin{pmatrix} B & * \\ * & * \end{pmatrix}

and

\textrm{max}\{\|M\|,\|N\|\} \leq C \times \textrm{max}\{\|A\|,\|B\|\}.

The equality (*) gives \| \alpha A + \beta B \| \leq \|\alpha M + \beta N\|, so we can get a bound on \|\alpha A + \beta B\| if we have a reasonable bound on M,N.

We are therefore led to the question: what is the best possible value of C?

I have worked on this problem with collaborators, and we have partial results. The optimal constant eludes us, and we are stuck, and we are not sure what the constant should be. What to do? We go back to option no. 1: experiment with examples.

The End

Ok. That is clearly more than seven minutes. To force myself to adhere to the seven minute limit, I made slides (there are four real slides there, according to the rule a slide for every two minutes).