Essential Spectra

Chapter 8 builds a toolkit for separating the essential spectrum from the discrete spectrum of Hilbert-space operators, motivated by problems in magnetohydrodynamics and topological insulators. After reviewing isolated eigenvalues, Riesz projections, and Fredholm theory, we compare the five standard notions of essential spectrum for non-normal operators and characterise each using singular sequences. We then introduce the essential injection modulus and show how to approximate it using modified singular values (e.g., higher singular values or the injection moduli of column-deleted truncations), yielding provably convergent algorithms for the essential spectrum, discrete spectrum, and multiplicities (algebraic and total discrete). Using the Solvability Complexity Index (SCI) hierarchy, we pinpoint the intrinsic difficulty of the associated computation and decision problems. Finally, the essential numerical range is studied as a diagnostic for spectral pollution: we compute it, derive error flags for finite-section methods, and prove lower bounds showing that certifying the absence of pollution can be harder than computing the spectrum itself.