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Computing spectra is a central problem in computational mathematics with anabundance of applications throughout the sciences. However, in manyapplications gaining an approximation of the spectrum is not enough. Often itis vital to determine geometric features of spectra such as Lebesgue measure,capacity or fractal dimensions, different types of spectral radii and numericalranges, or to detect essential spectral gaps and the corresponding failure ofthe finite section method. Despite new results on computing spectra and thesubstantial interest in these geometric problems, there remain no generalmethods able to compute such geometric features of spectra ofinfinite-dimensional operators. We provide the first algorithms for thecomputation of many of these longstanding problems (including the above). Asdemonstrated with computational examples, the new algorithms yield a library ofnew methods. Recent progress in computational spectral problems in infinitedimensions has led to the Solvability Complexity Index (SCI) hierarchy, whichclassifies the difficulty of computational problems. These results reveal thatinfinite-dimensional spectral problems yield an intricate infiniteclassification theory determining which spectral problems can be solved andwith which type of algorithm. This is very much related to S. Smale'scomprehensive program on the foundations of computational mathematics initiatedin the 1980s. We classify the computation of geometric features of spectra inthe SCI hierarchy, allowing us to precisely determine the boundaries of whatcomputers can achieve (in any model of computation) and prove that ouralgorithms are optimal. We also provide a new universal technique forestablishing lower bounds in the SCI hierarchy, which both greatly simplifiesprevious SCI arguments and allows new, formerly unattainable, classifications.

On the computation of geometric features of spectra of linear operators on Hilbert spaces