Special Lecture in Celebration of Marsha Berger and Margaret Wright

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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.