Colloquium: von Neumann's inequality on the disc and on the polydisc

Abstract: von Neumann's inequality provides a fundamental link between analytic functions on the unit disc and contraction matrices, or more generally contraction operators on Hilbert space. It asserts that if $T$ is a contraction and $p$ is a polynomial, then \[
   \|p(T)\| \le \sup \{ |p(z)| : |z| \le 1 \}.
\]
The multivariable setting turns out to be significantly more complicated. Whereas And\^o extended this inequality to pairs of commuting contractions, the corresponding statement for triples of commuting contractions is false. The first counterexamples were found in the early seventies, but this phenomenon is still not well understood and many questions remain open.

I will talk about the original inequality and some of the challenges in several variables. In particular, I will talk about the question of whether von Neumann's inequality for triples of commuting contractions holds up to a constant.

Host: John McCarthy

Reception to follow at Cupples I, Room 200 (Lounge) from 3:00 - 4:00 pm.