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

T0 review · 2 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper provides the first algorithms for computing geometric features of spectra—Lebesgue measure, capacity, box-counting and Hausdorff dimensions, spectral radii, and essential numerical ranges—and proves, via the Solvability…

desk verdict A substantial SCI-hierarchy paper with a real but repairable gap in the Lebesgue-measure lower-bound proof: the step-5 normal-operator construction puts all spectra on the imaginary axis, so the claimed optimality does not go through as written. read the letter →

arxiv 1908.09598 v5 pith:6DTWN6TY submitted 2019-08-26 math.SP

Matthew J. Colbrook This is my paper · ORCID
classification math.SP MSC 47A1046N4047A1247N5081Q1028A7828A12
keywords computationalspectralproblemsSolvabilityComplexityIndexhierarchyLebesguemeasureofspectrabox-countingdimensionHausdorffpollutioncapacityessentialnumericalrange
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper's central claim is that many long-unanswered "geometric" questions about spectra of bounded linear operators are computable, but only at a precisely calibrated cost: they need towers of algorithms with two, three, or even four nested limits, and no algorithm of the allowed type can get by with fewer. The quantities covered include the Lebesgue measure of the spectrum and pseudospectrum, whether that measure is zero, logarithmic capacity, box-counting and Hausdorff dimensions, spectral and essential spectral radii, the essential numerical range, and the decision problem of whether spectral pollution can occur in a given set. A sympathetic reader would care because these features control physically meaningful behaviour—wavepacket spreading, stability, gaps in essential spectra, and the failure of the finite-section method—and no general algorithms existed for them before. The proof combines new algorithms built on finite rectangular resolvent-norm approximations with a new lower-bound technique that equates the SCI hierarchy with the Baire hierarchy for certain combinatorial problems, so the impossibility results do not depend on a specific programming model.

What carries the argument

The load-bearing machinery is a pair of tools. The first is the finite rectangular resolvent approximation: the function $\gamma_n(z;A)=\min\{\sigma_1((A-zI)|_{P_nH}),\sigma_1((A^*-\bar z I)|_{P_nH})\}$, where $\sigma_1$ is the smallest singular value and $P_n$ the projection onto the first $n$ basis vectors, converges uniformly on compact sets from above to the reciprocal resolvent norm $\|R(z,A)\|^{-1}$. All the algorithms for measure, dimension, capacity and radii work by reading these approximations on a grid and taking nested limits as the truncation width, grid spacing, and resolvent cut-off go to zero; the $\Sigma$/ $\Pi$ classification encodes which direction the error is controlled in. The second is a new universal lower-bound technique: for the special combinatorial problems on $\{0,1\}^{\mathbb{N}}$ (deciding whether a 0-1 array has a column with infinitely many ones, and variants), the SCI hierarchy coincides exactly with the Baire hierarchy of descriptive set theory. Embedding these array problems into diagonal and block-operator spectral problems then proves that the corresponding spectral tasks cannot be solved by any general tower of lower height, regardless of the model of computation used.

What would settle it

Take the diagonal operator $D=\mathrm{diag}(r_1,r_2,\ldots)$ on $\ell^2(\mathbb{N})$ with $\{r_n\}$ a dense enumeration of $[0,1]$, so $Leb(Sp(D))=1$. Any algorithm that reads finitely many entries of $D$ must give the same output on $D$ as on some finite diagonal operator whose spectrum is a finite set (measure zero), so its output cannot eventually lie within $1/2$ of both answers; running the paper's LebSpec tower on this $D$ and on its finite truncations is a concrete experiment that exhibits the two-limit convergence and the failure of any one-limit method.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is a sharp classification, not just a collection of algorithms. For general bounded operators with the basic evaluation set $\Lambda_1$ (finitely many matrix entries per step), the Lebesgue measure of the spectrum is computable by a $\Pi^A_3$ tower and is not computable by any $\Delta^G_3$ tower (Theorem 3.14); the measure of pseudospectra is easier, $\Sigma^A_2$ with $\Lambda_1$ and $\Sigma^A_1$ with $\Lambda_2$ (Theorem 3.15). Deciding whether the spectrum has Lebesgue measure zero is strictly harder still, $\Pi^A_4$ for self-adjoint operators with $\Lambda_1$ (Theorem 3.18)—the paper's first spectral problems requiring four limits. For self-adjoint operators, box-counting dimension of the spectrum is $\Pi^A_3$ and Hausdorff dimension is $\Sigma^A_4$ when only $\Lambda_1$ is available, dropping by one limit when $\Lambda_2$ is allowed (Theorem 3.20). The paper also shows that computing the spectral radius of a general operator is exactly as hard as computing the spectrum itself ($\Pi^A_3$), and that detecting whether spectral pollution can occur on a set is strictly harder than the spectral computation the finite-section method was designed to solve. These classifications are accompanied by explicit algorithms, pseudocode, and computational demonstrations on operators such as the almost Mathieu operator and a Laplacian on a Penrose tiling.

Load-bearing premise

All the impossibility results are proved for algorithms whose only information about an operator is finitely many matrix entries per step (or, with $\Lambda_2$, finitely many entries of $A^*A$ and $AA^*$ as well); if a different oracle were allowed—say, direct queries to the resolvent norm—the number of required limits could drop.

Editorial extensions

If this is right

  • Computing the spectral radius of a general bounded operator is no easier than computing the spectrum itself, contradicting the naive expectation from Gelfand's formula that one limit suffices.
  • For self-adjoint operators, the Lebesgue measure of the pseudospectrum can be approximated with one or two limits (depending on the evaluation set), and letting the pseudospectral radius tend to zero gives the measure of the true spectrum; deciding whether the spectrum is Lebesgue null requires up to four limits.
  • Box-counting dimension of self-adjoint spectra is computable with a $\Pi^A_3$ tower, and Hausdorff dimension with a $\Sigma^A_4$ tower, when only matrix entries are read; the extra limit reflects the countable stability of Hausdorff dimension.
  • Detecting a gap in the essential spectrum, equivalently deciding whether spectral pollution can occur on a given set, is strictly harder than the spectral computation itself ($\Sigma^A_3$), so a 'failure flag' for the finite-section method cannot be produced by the same algorithm that computes the spectrum.
  • The new lower-bound technique extends the SCI hierarchy beyond height three; it gives the first spectral decision problems with SCI exactly four, and simplifies earlier proofs of lower bounds.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the information model were enriched—for example, by allowing algorithms to query the resolvent norm directly, or to read entries selected by a non-computable rule—the classifications in this paper would not necessarily survive; the advertised universality is universality over models of computation that share the $\Lambda_1$/ $\Lambda_2$ evaluation sets.
  • Because the lower bounds rest on the equivalence with the Baire hierarchy, the same combinatorial embeddings could be applied to other high-SCI problems outside spectral theory, such as computing invariant measures, attractors, or solution manifolds of PDEs, whenever the problem can be reduced to deciding properties of infinite 0-1 arrays.
  • The algorithms' monotone convergence makes them suitable for rigorous computer-assisted proofs: running the $\Pi^A_k$ towers with interval arithmetic can certify upper bounds on measures and dimensions of spectra, which is exactly the kind of certificate needed to confirm conjectures for quasicrystal or random Schrödinger operators beyond the one-dimensional almost Mathieu case.
  • The Penrose-tiling computations suggest a testable physical prediction: the part of the spectrum above $-3$ for the graph Laplacian on a Penrose tiling has Lebesgue measure zero and box-counting dimension near $0.8$; a future rigorous implementation of the same algorithm could turn this numerical evidence into a theorem.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 3 minor

Summary. The paper develops a systematic Solvability Complexity Index (SCI) classification for the computation of geometric features of spectra of bounded linear operators on Hilbert space. The quantities treated include the spectral radius and essential spectral radius, polynomial operator norms and logarithmic capacity, the essential numerical range, detection of spectral pollution and essential spectral gaps, the Lebesgue measure of spectra and pseudospectra, the property of having Lebesgue-null spectrum, and box-counting and Hausdorff dimensions of spectra. For each problem the paper proposes explicit towers of algorithms, often with pseudocode, and proves lower bounds by reduction to combinatorial problems that are shown to be complete at various levels of the Baire hierarchy via a new general technique (Theorem 5.19). The headline theorems are the classifications in Theorems 3.3, 3.5, 3.6, 3.10, 3.14, 3.15, 3.18 and 3.20, summarized in Table 1.

Significance. If the results stand, this is a substantial contribution to the foundations of computational spectral theory. The paper provides the first SCI-sharp algorithms for several longstanding geometric spectral quantities, and the new Baire-hierarchy reduction technique is a genuine methodological advance that both simplifies earlier lower-bound proofs and reaches beyond SCI level 3. The upper-bound constructions are detailed, the pseudocode is concrete, and the computational examples (almost Mathieu operator, Penrose-tile Laplacian, non-normal spectral radius, essential numerical range) illustrate that the algorithms are not merely existence proofs. The self-adjoint and diagonal-operator classifications appear carefully proved. The main caveat is a specific gap in the lower-bound proof for Lebesgue measure for general normal and general bounded operators, which propagates to several related optimality claims; this is a load-bearing issue for the advertised sharpness in those cases, although the upper bounds and the self-adjoint cases are not affected.

major comments (2)
  1. [Section 8, proof of Theorem 3.14, Step 5] The claimed lower bound for Lebesgue measure for Ω = ΩB, ΩN, Ωg with the evaluation set Λ1 is not established as written. The construction replaces each C(j) by D(j) = ⊕_{k=1}^j i h_k C(j), where h_k is dense in [0,1]. Since C(j) is real symmetric with Sp(C(j)) ⊂ [-1,1] by Lemma 6.4, each i h_k C(j) has spectrum contained in the purely imaginary line segment i[-h_k,h_k]. Consequently the spectrum of the direct sum is a countable union of finite sets and has two-dimensional Lebesgue measure zero regardless of the encoded array. The asserted dichotomy 'positive two-dimensional Lebesgue measure iff Ξ̃2 = 0' therefore fails, and the reduction to the SCI = 3 combinatorial problem yields no contradiction. This invalidates the lower bound {Ξ^L_1, Ω, Λ1} ∉ Δ^G_3 for Ω = ΩB, ΩN, Ωg as written. The same gap propagates to the lower bounds of Theorem 3.15, which are derived directly from Theorem 3.14, and to the Λ1 lower bounds for ΩB, ΩN, Ωg in Theorem 3.18, which the proof says follow from the same Step-5 construction. A repair requires a genuinely two-dimensional spectral embedding, for example a normal diagonal operator whose eigenvalues accumulate on a set of positive area; replacing h_k by points in the unit disk does not suffice, since each summand still has one-dimensional spectrum.
  2. [Section 3.5, remark after Theorem 3.20] The remark states that the Hausdorff-dimension proofs 'can be adapted with an additional limit and the use of two-dimensional covering boxes to treat the class of general bounded operators', but no proof or even sketch is provided and the details are explicitly omitted. This is an unproved classification claim. The main theorem is stated only for self-adjoint operators, so this extension is not load-bearing for the central results, but it should either be proved in an appendix or clearly marked as outside the scope of the paper.
minor comments (3)
  1. [Section 3.2, before Theorem 3.3] The text 'easier than the general clmss ΩB' contains a typo: 'clmss' should be 'class'.
  2. [Appendix B, Algorithm 12 (NullLebSpec)] The pseudocode for NullLebSpec calls LebPseudoSpec(n1, n2, f(n1), cn1, A), but the signature of LebPseudoSpec in Algorithm 11 is LebPseudoSpec(n, A, ϵ). The arguments do not match, and no value of ϵ is supplied. Also, the loop 'for j = 1,...,n1' computes the identical quantity each time, so either the loop index should affect the computation or the loop should be removed.
  3. [Theorem 3.14 proof, Step 1] In the definition of U(n1,n2,A), the function Fn1(z) is used but the surrounding text sometimes writes Fn(z); this is understandable but should be made uniform to avoid confusion between the first and second limit parameters.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the new geometric-feature classifications are derived from resolvent-norm approximations and external descriptive set theory, not from their own conclusions.

full rationale

The paper's central algorithms for Lebesgue measure, capacity, box-counting dimension and Hausdorff dimension are built from convergent approximations of the resolvent norm (the gamma functions and DistSpec routines from prior work [49,51]) and from combinatorial embedding problems whose hardness is established through the Baire hierarchy. The prior resolvent-norm results are reused as building blocks, but their stated assumptions do not include the geometric quantities being computed, so the derivation is not equivalent to its inputs. The lower bounds are reductions from array problems in Theorem 5.19, whose hardness is proved via Wadge-completeness and the Lebesgue-Hausdorff-Banach theorem, both external to the paper's conclusions. No fitted parameter is renamed as a prediction, and no uniqueness theorem from the author's prior work is invoked to force a choice. The paper's heavy reliance on earlier SCI papers [20,49,51] is self-citation, but it is not circular in the load-bearing sense because those papers give independent results with stated assumptions that do not presuppose the new geometric-feature theorems. The Step 5 construction in Theorem 3.14 noted in the review contains a possible spectral-lemma defect, but that is a correctness or proof gap rather than circularity: even if that reduction fails, the claimed classification is not being assumed as an input. Accordingly, no circular step is identified.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new free parameters and no invented physical/mathematical entities. Its central results rest on (i) standard descriptive set theory, (ii) the SCI hierarchy framework with general algorithms and evaluation sets, and (iii) known resolvent-norm approximation theorems from previous papers, several by the same author. These are external benchmarks rather than assumptions of the target result, so the circularity burden is low.

assumptions (4)
  • standard math ZFC set theory and standard descriptive set theory: Lebesgue-Hausdorff-Banach theorem and Wadge's theorem for Borel sets in zero-dimensional spaces
    Used in Section 5.3 and 5.4 to prove Theorems 5.17 and 5.19 (SCI vs Baire hierarchy). These are cited standard results, not proved in the paper.
  • domain assumption Definition of a general algorithm and arithmetic tower (Definition 5.1-5.3): finite adaptive information reads from the evaluation set; recursive/BSS operations for arithmetic towers
    Defines the SCI hierarchy. Every classification and optimality claim is relative to this framework.
  • domain assumption Operator classes Ω_D, Ω_N, Ω_SA, Ω_f, Ω_g and evaluation sets Λ1/Λ2 as defined in Section 3.1
    The results are stated for these classes; the bounds on dispersion (c_n) and resolvent (g) are assumed known for the respective classes.
  • standard math Uniform convergence of the finite-section resolvent norm approximations γ_{n,m}, γ_n and DistSpec to ||R(z,A)||^{-1} on compact subsets (Theorem from [51,84])
    Foundational for the algorithms in Sections 8 and 9; the paper quotes these convergence results and does not reprove them.

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Pith. "Pith review of On the computation of geometric features of spectra of linear operators on Hilbert spaces." pith.science (2026). https://pith.science/paper/6DTWN6TY

@misc{pith2026190809598,
  author       = {Pith},
  title        = {Pith review of: On the computation of geometric features of spectra of linear operators on Hilbert spaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6DTWN6TY}},
  note         = {Machine review of arXiv:1908.09598}
}
read the original abstract

Computing spectra is a central problem in computational mathematics with an abundance of applications throughout the sciences. However, in many applications gaining an approximation of the spectrum is not enough. Often it is vital to determine geometric features of spectra such as Lebesgue measure, capacity or fractal dimensions, different types of spectral radii and numerical ranges, or to detect essential spectral gaps and the corresponding failure of the finite section method. Despite new results on computing spectra and the substantial interest in these geometric problems, there remain no general methods able to compute such geometric features of spectra of infinite-dimensional operators. We provide the first algorithms for the computation of many of these longstanding problems (including the above). As demonstrated with computational examples, the new algorithms yield a library of new methods. Recent progress in computational spectral problems in infinite dimensions has led to the Solvability Complexity Index (SCI) hierarchy, which classifies the difficulty of computational problems. These results reveal that infinite-dimensional spectral problems yield an intricate infinite classification theory determining which spectral problems can be solved and with which type of algorithm. This is very much related to S. Smale's comprehensive program on the foundations of computational mathematics initiated in the 1980s. We classify the computation of geometric features of spectra in the SCI hierarchy, allowing us to precisely determine the boundaries of what computers can achieve (in any model of computation) and prove that our algorithms are optimal. We also provide a new universal technique for establishing lower bounds in the SCI hierarchy, which both greatly simplifies previous SCI arguments and allows new, formerly unattainable, classifications.

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Cited by 2 Pith papers

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