Spectral Sets

TL;DR

This survey reviews spectral sets, including K-spectral sets, and their role in estimating matrix function norms, highlighting their importance across various mathematical and applied fields.

Contribution

It provides a comprehensive overview of spectral sets, their properties, examples, and applications, connecting linear algebra, operator theory, and complex analysis.

Findings

Spectral sets help estimate norms of matrix functions.

Examples include numerical range and pseudospectrum.

Applications span numerical linear algebra and differential equations.

Abstract

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.