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NOTE-009: Spectral Sets

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PaperLIT-011 — Spectral Sets

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Contribution

The chapter is a structured survey, not a research paper. It collects in handbook format (Definitions / Facts / Examples / Open Problems) the modern theory of spectral and K-spectral sets for Hilbert-space operators and matrices:

  • basic properties and characterisations (§107.2);
  • variations of von Neumann’s inequality, including Banach-space and constrained versions (§107.3);
  • multivariable von Neumann inequalities and their failure for n ≥ 3 (§107.4);
  • dilations, complete bounds and similarity to contractions (§107.5);
  • intersections of (K-)spectral sets and the annulus problem (§107.6);
  • the numerical range as a K-spectral set (§107.7);
  • applications to computing and bounding matrix functions and Krylov methods (§107.8).

The only result it presents as new is the improved annulus lower bound in Example 107.6.2.

Key insight

For a normal matrix, ‖f(A)‖ = max over the spectrum of |f|. For a non-normal matrix the spectrum is the wrong set: it is not even K-spectral unless A is diagonalisable, and then K is the eigenvector condition number. The right move is to enlarge the set until the constant is controlled. The numerical range W(A) is the canonical choice. It is always K-spectral with a universal K ≤ 11.08 (conjecturally 2), so any polynomial or rational approximation error of f on a convex set E ⊃ W(A) transfers to the matrix at the cost of at most that constant (§107.8).

Assumptions

  • Setting:
    • A is a bounded operator on a complex Hilbert space H, or a matrix on ℂⁿ with the Euclidean norm, unless a Fact says otherwise;
    • X is closed with σ(A) ⊂ X;
    • f ranges over rational functions with poles off X (R(X)), and f(A) is defined by the Riesz functional calculus (§107.2).
  • Hilbert-space dependence: von Neumann’s inequality characterises Hilbert spaces (Fact 107.3.1). Banach-space versions carry worse constants: 3 for Banach contractions (Fact 107.2.12), and (πn+1) for subordinate matrix norms (Fact 107.3.3).
  • §107.8:
    • E is compact and convex, K(E)-spectral for A, and f is analytic on E;
    • the Faber-based bounds need the conformal map φ of the exterior of E;
    • the Padé bounds are for Markov functions f(z) = c + ∫ dμ(x)/(z−x) with β < −w(A), and some also need E symmetric about ℝ.

Key results

Theorem-level entries as the chapter states them, all attributed to the literature.

  • Disk and half-plane von Neumann inequalities (Fact 107.2.1):
    • {|z−α| ≤ r} is spectral iff ‖A−αI‖ ≤ r;
    • a closed half-plane is spectral iff it contains W(A).
  • 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.
  • Pseudospectrum (Example 107.2.7): the closure of the ε-pseudospectrum is K-spectral with K = length(∂X)/(2πε).
  • Numerical-radius disk (Example 107.2.8): {|z| ≤ w(A)} is 2-spectral, and 2 is best possible.
  • Class C_s (Fact 107.2.10): the unit disk is K-spectral for A ∈ C_s, with best constant K = max(1, s) ([OA75]).
  • Banach contractions (Fact 107.2.12): ‖p(A)‖ ≤ sup_{|z|≤3}|p(z)|, and 3 is sharp.
  • Bohr (Fact 107.3.4a): ‖p(rS)‖_{L(ℓ¹)} ≤ ‖p‖_D iff r ≤ 1/3.
  • Multivariable (§107.4):
    • holds for commuting pairs (Andô) and for Σ‖A_j‖² ≤ 1 (Brehmer);
    • fails for n = 3 with 5×5 matrices (Varopoulos, Example 107.4.2);
    • explicit 8×8 counterexample with ‖p(A)‖ ≥ 4 > ‖p‖_{𝕋³} (Crabb–Davie, Example 107.4.3);
    • whether a finite C_n exists is open (Dixon), with C_n growing faster than any power of n if it does.
  • Dilation and similarity (§107.5):
    • Arveson: a normal ∂X-dilation exists iff X is completely spectral;
    • Paulsen: A is similar to a contraction iff D is completely K-spectral;
    • Pisier: a polynomially bounded operator not similar to a contraction;
    • Peller/Bourgain: for M = sup‖Aⁿ‖, ‖p(A)‖ ≤ M²(log d)‖p‖_D;
    • Bourgain: a matrix with ‖f(A)‖ ≤ C‖f‖_D is similar to a contraction with ‖L⁻¹‖‖L‖ ≤ K C⁴ log(n+1).
  • Intersections of spectral disks (Fact 107.6.4, [BBC09]):
    • n spectral disks intersect in a complete K-spectral set with K ≤ n + n(n−1)/√3;
    • the convex case gives 11.08;
    • for n = 2, K = 2 + 2/√3.
  • Annulus X(R) = {R⁻¹ ≤ |z| ≤ R}, for ‖A‖, ‖A⁻¹‖ ≤ R (Fact 107.6.5):
    • Shields: K(R) ≤ 2 + √((R²+1)/(R²−1));
    • [BBC09]: K_cb(R) ≤ 2 + (R+1)/√(R²+R+1) ≤ 2 + 2/√3;
    • K(R) ≤ 3 for R ≥ 2.0953;
    • Crouzeix 2012 gives an integral bound;
    • lower bound: K(R) ≥ 4/3 ([BBC09]), improved in Example 107.6.2 to > 3/2.
  • Numerical range, shape-dependent constants (Fact 107.7.1): every compact convex X ⊃ W(A) is completely K-spectral with
    • K = 3 + (2π diam(X)²/area(X))³ (Delyon–Delyon 1999);
    • K = 1 + 2/(1−q(X)) (Putinar–Sandberg), where q(X) is Neumann’s configuration constant;
    • K = 2 + π + TV(log r) (Badea–Crouzeix–Delyon).
  • Numerical range, universal constant (Fact 107.7.2a, [Cro07, Theorem 1]): K_Crouzeix ∈ [2, 11.08] such that the closure of W(A) is complete K-spectral for every bounded A. For 2×2 matrices, W(A) is completely 2-spectral (Fact 107.7.2b).
  • Numerical range, special shapes (Example 107.7.3):
    • sector, strip, parabola and hyperbola: 2 + 2/√3;
    • ellipse with eccentricity e: 2 + 2/√(4−e²);
    • equilateral triangle: 2 + π + 6 log 2;
    • square: 2 + π + 4 log 2.
  • Drury (Fact 107.7.3d): w(A) ≤ 1 implies w(f(A)) ≤ (5/4)‖f‖_D, and 5/4 is best.
  • Crouzeix’s conjecture: known to hold for generalised Jordan blocks with a scalar in the lower-left entry (Example 107.7.5, [GC12]).
  • Applications (§107.8):
    • Faber-sum error: ‖(f − p_m)(A)‖ ≤ 2 Σ_{j>m} |f_j| ([BR09]), since the Faber operator satisfies ‖F(F)(A)‖ ≤ 2 max_D|F| ([Bec05]);
    • Arnoldi f(A)b error: ε_m ≤ 2K_Crouzeix ρ_{m−1}(f,E) or ε_m ≤ 4 Σ_{j≥m}|f_j|;
    • GMRES residual: ‖r_m‖/‖r_0‖ ≤ min{1, 2/|F_m(0)|} ≤ (2 + 1/|φ(0)|)/|φ(0)|
    • for positive-definite A + A* (lens E), the convergence factor is 1/|φ(0)| = 2 sin(β/(4−2β/π)) < sin β, with cos β = dist(0, W(A))/w(A);
    • FOM error: ≤ 4|φ(0)|^{−m}/dist(0, E).

Claims

idclaimstrengthsupport
C1σ(A) is K-spectral iff A is diagonalisable (K = eigenvector condition number)strongproof sketched in Examples 107.2.3–107.2.5
C2A is normal / Hermitian / unitary iff σ(A) / ℝ / 𝕋 is spectralstrongconverse proved in Example 107.2.6 via Möbius maps and Lagrange interpolation
C3The closure of W(A) is a complete K-spectral set with a universal K ∈ [2, 11.08]strong (as report)cites [Cro07, Thm 1]; no proof in the chapter
C4For 2×2 matrices, K = 2 is attained and suffices (completely)strong (as report)cites [Cro04a, Thm 1.1]; Example 107.7.7 shows 2 is attained
C5Spectral and completely spectral constants for W(A) can differ (3×3 example)weak–moderateExample 107.7.4, sourced to “personal communication” [Cro12b]; no proof given
C6The annulus X(R) is not 3/2-spectral for the explicit 2×2 A, for any R > 1strongexplicit computation in Example 107.6.2; checkable by hand
C7The multivariable von Neumann inequality fails for n = 3strongexplicit counterexamples (Examples 107.4.2–107.4.3), from [Var74, CD75]
C8Arnoldi and GMRES errors are bounded by polynomial/Faber approximation errors on E ⊃ W(A), times at most 2K_Crouzeix or 4strong (as report)[Bec05, BR09]
C9Is K_Crouzeix = 2 (even for 3×3)?open problemOpen Problem 107.7.1

Concepts

  • (K-)spectral set — a closed X ⊃ σ(A) with ‖f(A)‖ ≤ K‖f‖_X for all f ∈ R(X). “Spectral” means K = 1.
  • Complete K-spectral set — the same inequality for all matrices (f_ij) of rational functions, in the operator norm on ⊕ⁿH. It is equivalent to X being completely spectral for some B similar to A with ‖L⁻¹‖‖L‖ ≤ K (Fact 107.5.2).
  • Numerical range / numerical radius — W(A) = {⟨Ax,x⟩ : ‖x‖ = 1} and w(A) = sup|W(A)|, with ‖A‖/2 ≤ w(A) ≤ ‖A‖.
  • Class C_s (Sz.-Nagy–Foias) — operators with a unitary s-dilation. C₁ is the contractions; C₂ is the numerical-radius contractions.
  • Normal ∂X-dilation — A = P_H f(N)|_H with N normal and σ(N) ⊂ ∂X.
  • Faber polynomials / Faber operator — the polynomial parts of φᵐ at ∞, for φ the exterior conformal map of E. The operator carries F on D to Σ f_j F_j.

Connections

The chapter:

  • builds on von Neumann’s 1951 paper;
  • organises the Sz.-Nagy–Foias dilation theory, the Arveson/Paulsen complete-boundedness framework and the Pisier counterexample around the single notion of a K-spectral set;
  • ties the numerical-range line (Delyon–Delyon 1999 → Crouzeix 2004/2007 → the Badea–Beckermann–Crouzeix lens/sector/annulus work) to Krylov-method error analysis (Beckermann 2005; Beckermann–Reichel 2009).

Both authors are among the contributors to the results it reports (BBC09, BC02, BC06, BC07, Bec05, BR09).

Bearing on the record

No THEORY document in this record rests on it, and it contradicts none. It is a reference for any future note that needs a norm bound on a function of a non-normal matrix.

For ML practice it carries no instruction, and the chapter mentions no ML. The anthology practice closest in subject is ANTH-SOTA-011: extreme Hessian eigenvalues by Lanczos on Hessian-vector products. It works with a symmetric Hessian, which is normal, so σ(A) is already a spectral set (Fact 107.2.6) and the chapter’s non-normal machinery adds nothing there. Where the chapter would matter is analysing non-normal linear recurrences or Krylov approximations of f(A)b for non-symmetric A. Those settings would call for its §107.7–107.8 bounds; no anthology document currently makes such a claim.

Limitations

  • A survey with almost no proofs. Apart from the short arguments in §107.2’s Examples and the Example 107.6.2 computation, every Fact is cited, and its support is the cited paper.
  • One entry rests on unpublished support. Example 107.7.4 is sourced to a personal communication.
  • Frozen at early 2013. In particular, the upper bound 11.08 in Fact 107.7.2a is no longer the best known. I recall Crouzeix–Palencia (2017) proving 1 + √2, but did not verify it in this session. Whether Crouzeix’s conjecture K = 2 has since been settled was not checked.
  • Uneven scope of the application bounds. The §107.8 bounds need E convex and containing W(A). When W(A) contains or nearly contains 0, which can happen for indefinite or highly non-normal A, the GMRES and FOM bounds become vacuous: 1/|φ(0)| → 1.

Open questions

As listed in the chapter, with what would close each.

  • Is K_Crouzeix = 2, even for 3×3 matrices? (Open Problem 107.7.1.) Closed by a proof of the 2 bound, or a matrix exceeding it.
  • Does a finite C_n exist with ‖f(A₁,…,A_n)‖ ≤ C_n‖f‖_{D^n} for all commuting contractions? (Open Problem 107.4.1, Dixon.) It is believed not.
  • What obstructs commuting unitary dilations of commuting contraction n-tuples? (Open Problem 107.5.1.)
  • Is the intersection of two K-spectral sets K′-spectral in general? (§107.6 preamble.) It is known when the boundaries are disjoint (Fact 107.6.1).
  • What is the optimal annulus constant K(R)? Currently between > 3/2 (Example 107.6.2) and the upper bounds of Fact 107.6.5.

Corrections to the seeded skim

  • Several dossier locators are wrong: most of the §107.2 items it cites are Examples, not Facts.
    • “σ(A) is K-spectral iff A is diagonalisable” is Example 107.2.5, with the proof sketched there. It is not “Facts 5”. Fact 107.2.5 is Williams’ existence of minimal spectral sets.
    • The characterisations of normal, Hermitian and unitary A by σ(A), ℝ and 𝕋 being spectral are Fact 107.2.9(a)–(c), with the converse proved in Example 107.2.6. Fact 107.2.6 is only von Neumann’s “A normal ⇒ σ(A) is a (minimal) spectral set”.
    • The pseudospectrum result is Example 107.2.7, K = length(∂X)/(2πε), cross-referred to Fact 16.3.5 of the Handbook. It is not Fact 7. Fact 107.2.7 is Williams’ criterion using functions vanishing at an interior point.
  • The dossier’s definition drops a condition. K-spectrality also requires σ(A) ⊂ X (§107.2, Definitions), not just the norm inequality.
  • The chapter contains one result it presents as new, which the dossier misses. Example 107.6.2 gives the lower bound K(R) ≥ 2(1+R²+R)/(1+R²+2R) > 3/2 for the annulus X(R) = {R⁻¹ ≤ |z| ≤ R}. It uses f(z) = g(z) − g(1/z), g(z) = R(z−1)/(R²−z), on A = [[1, R−1/R],[0,1]]. The chapter notes that this improves the lower bound K(R) ≥ 4/3 of Fact 107.6.5c ([BBC09]). The computation is explicit and checkable.
  • The constant 11.08 appears twice, for two different results. Fact 107.7.2a (Crouzeix 2007) gives it for the numerical range. Fact 107.6.4a gives it for convex intersections of spectral disks. The dossier cites only the first.
  • Spectral and completely spectral constants can differ. Example 107.7.4 (Crouzeix, personal communication) gives a 3×3 matrix whose W(A) is 2-spectral but not completely 2-spectral. This limits how far the 2×2 result (Fact 107.7.2b) can be read as evidence for K = 2 in the complete sense. The dossier’s “2×2 matrices are completely 2-spectral” is correct but omits this example.
  • The dossier’s section list, Delyon–Delyon, the Crouzeix interval [2, 11.08] and the §107.8 applications check out. The dossier’s “ML link (SSM/RNN transition matrices, Krylov)” is its own inference; the chapter makes no such connection.

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