Chapter 37
Spectral Sets
Catalin Badea
Université Lille 1
Bernhard Beckermann
Université Lille 1
Spectral sets and K-spectral sets, introduced by John von Neumann in [vNe51], 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 23 and 25. Estimating the norm of functions of matrices is an essential task in numerous fields of pure and applied mathematics, such as (numerical) linear algebra [Gre97, Hig08], functional analysis [Pau02], and numerical analysis. More specific examples include probability [DD99], semi-groups and existence results for operator-valued differential equations, ...
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