  We use results in [M. Crouzeix and A. Greenbaum,Spectral sets: numerical
range and beyond, SIAM Jour. Matrix Anal. Appl., 40 (2019), pp. 1087-1101] to
derive a variety of K-spectral sets and show how they can be used in some
applications. We compare the K-values derived here to those that can be derived
from a straightforward application of the Cauchy integral formula, by replacing
the norm of the integral by the integral of the resolvent norm. While, in some
cases, the new upper bounds on the optimal K-value are much tighter than those
from the Cauchy integral formula, we show that in many cases of interest, the
two values are of the same order of magnitude, with the bounds from the Cauchy
integral formula actually being slightly smaller. We give a partial explanation
of this in terms of the numerical range of the resolvent at points near an
ill-conditioned eigenvalue.
