14:00
Let $A$ be an $n$ by $n$ matrix or a bounded linear operator on a complex Hilbert space $(H, \langle \cdot , \cdot \rangle , \| \cdot \|)$. A closed set $\Omega \subset \mathbb{C}$ is a $K$-spectral set for $A$ if the spectrum of $A$ is contained in $\Omega$ and if, for all rational functions $f$ bounded in $\Omega$, the following inequality holds:
\[\| f(A) \| \leq K \| f \|_{\Omega} ,\]
where $\| \cdot \|$ on the left denotes the norm in $H$ and $\| \cdot \|_{\Omega}$ on the right denotes the $\infty$-norm on $\Omega$. A simple way to obtain a $K$ value for a given set $\Omega$ is to use the Cauchy integral formula and replace the norm of the integral by the integral of the resolvent norm:
\[f(A) = \frac{1}{2 \pi i} \int_{\partial \Omega} ( \zeta I - A )^{-1}
f( \zeta )\,d \zeta \Rightarrow
\| f(A) \| \leq \frac{1}{2 \pi} \left( \int_{\partial \Omega}
\| ( \zeta I - A )^{-1} \|~| d \zeta | \right) \| f \|_{\Omega} .\]
Thus one can always take
\[K = \frac{1}{2 \pi} \int_{\partial \Omega} \| ( \zeta I - A )^{-1} \| | d \zeta | .\]
In M. Crouzeix and A. Greenbaum, Spectral sets: numerical range and beyond, SIAM J. Matrix Anal. Appl., 40 (2019), pp. 1087-1101, different bounds on $K$ were derived. I will show how these compare to that from the Cauchy integral formula for a variety of applications. In case $A$ is a matrix and $\Omega$ is simply connected, we can numerically compute what we believe to be the optimal value for $K$ (and, at least, is a lower bound on $K$). I will show how these values compare with the proven bounds as well.
(joint with Michel Crouzeix and Natalie Wellen)
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