A set $\Omega$ is a spectral set for an operator $T$ if the spectrum of $T$ is contained in $\Omega$, and von Neumannâ€™s inequality holds for $T$ with respect to the algebra $R(\Omega)$ of rational functions with poles off of the closure of $\Omega$. It is a complete spectral set if for all $n \in \mathbb{N}$, the same is true for $M_n(\mathbb C) \otimes R(\Omega)$. The rational dilationâ€‚problem asks, if $\Omega$ is a spectral set for $T$, is it a complete spectral set for $T$? There are natural multivariable versions of this. There are a few cases where rational dilation is known to hold (e.g., over the disk and bidisk), and some where it is known to fail, for example over the Neil parabola, a distinguished variety in the bidisk. The Neil parabola is an example of a variety naturally associated to a constrained subalgebra of the disk algebra, namely $\mathbb{C} + z^2 A(\mathbb D)$. This talk discusses why rational dilation fails for a large class of such varieties associated to constrained algebras.

- All seminars.
- Seminars for 2024

Last updated: 15 Jul 2024