Noncommutative Analysis

Month: August, 2015

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!

 

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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.