  This is a survey about spectral sets, to appear in the second edition of
Handbook of Linear Algebra (L. Hogben, ed.). Spectral sets and K-spectral sets,
introduced by John von Neumann, offer a possibility to estimate the norm of
functions of matrices in terms of the sup-norm of the function. Examples of
such spectral sets include the numerical range or the pseudospectrum of a
matrix, discussed in Chapters 16 and 18. Estimating the norm of functions of
matrices is an essential task in numerous fields of pure and applied
mathematics, such as (numerical) linear algebra, functional analysis, and
numerical analysis. More specific examples include probability, semi-groups and
existence results for operator-valued differential equations, the study of
numerical schemes for the time discretization of evolution equations, or the
convergence rate of GMRES (Section 41.7). The notion of spectral sets involves
many deep connections between linear algebra, operator theory, approximation
theory, and complex analysis.
