Source
Badea & Beckermann (2013), “Spectral Sets”, chapter 107 of the Handbook of Linear Algebra, 2nd ed., LIT-011, read in full in NOTE-009.
What was actually shown
A closed set X containing σ(A) is K-spectral for A when ‖f(A)‖ ≤ K·sup_X|f| for every rational f with poles off X (§107.2). For a normal matrix the spectrum is spectral with K = 1. The chapter collects what holds when A is not normal.
The spectrum (Example 107.2.5). σ(A) is K-spectral iff A is diagonalisable, with K ≤ cond(V) for V the eigenvector matrix. No finite set is K-spectral for a non-diagonalisable matrix. The proof is sketched in Examples 107.2.3–107.2.5, and the reading rates it strong (NOTE-009, C1). The characterisations of normal, Hermitian and unitary A by σ(A), ℝ or 𝕋 being spectral are Fact 107.2.9, with the converse in Example 107.2.6 (C2).
The numerical range (Fact 107.7.2). For every bounded operator A on a Hilbert space, the closure of W(A) = {⟨Ax, x⟩ : ‖x‖ = 1} is a complete K-spectral set with a universal K_Crouzeix ∈ [2, 11.08] ([Cro07, Thm 1]). For 2×2 matrices W(A) is completely 2-spectral, and 2 is attained (Fact 107.7.2b, Example 107.7.7). Shape-dependent constants exist for any compact convex X ⊃ W(A) (Fact 107.7.1), and special shapes have explicit ones, such as 2 + 2/√3 for sectors, strips, parabolas and hyperbolas (Example 107.7.3).
Other sets. The closure of the ε-pseudospectrum is K-spectral with K = length(∂X)/(2πε) (Example 107.2.7). The numerical-radius disk is 2-spectral, and 2 is best possible (Example 107.2.8).
What follows for computation (§107.8). Any polynomial or rational approximation error of f on a convex E ⊃ W(A) transfers to the matrix at a cost of at most that constant. For Arnoldi, ε_m ≤ 2K_Crouzeix ρ_{m−1}(f, E).
What could have come out otherwise. The spectrum could have controlled ‖f(A)‖ for some non-diagonalisable A, and the numerical range’s constant could have grown with the dimension. Neither is the case.
What this does not say
- That the constant is 2. Whether K_Crouzeix = 2, even for 3×3 matrices, is Open Problem 107.7.1. The 2×2 case is weak evidence for the complete version, since Example 107.7.4 gives a 3×3 matrix whose W(A) is 2-spectral but not completely 2-spectral. That example rests on a personal communication (NOTE-009, C5).
- That 11.08 is current. The chapter is frozen at early 2013. NOTE-009 recalls Crouzeix and Palencia (2017) proving 1 + √2, from outside the chapter and unverified. Check it before quoting either number as the best known.
- That the chapter proves these results. Apart from §107.2’s short arguments and the annulus computation of Example 107.6.2, every Fact is cited, and its support is the cited paper.
- That the Krylov bounds are always informative. They need E convex and containing W(A). When W(A) contains or nearly contains 0, the GMRES and FOM bounds become vacuous (NOTE-009, Limitations).
- Anything about learning systems. The chapter mentions no ML. For a symmetric Hessian, which is normal, the spectrum is already spectral and this machinery adds nothing.
Connections
- THEORY-325 is the dynamical face of the same gap: a non-normal linear system can be destabilised by time dependence while stable at every instant, and non-normality is necessary for it. There the spectrum fails to control a product of operators; here it fails to control a function of one.