Generalised eigenvector expansion of infinite Toeplitz matrices with absolutely/completely monotone entries

Jacek Wszo{\l}a; Mateusz Kwa\'snicki

arxiv: 2606.01986 · v1 · pith:UXP2E2VMnew · submitted 2026-06-01 · 🧮 math.SP · math.PR

Generalised eigenvector expansion of infinite Toeplitz matrices with absolutely/completely monotone entries

Mateusz Kwa\'snicki , Jacek Wszo{\l}a This is my paper

Pith reviewed 2026-06-28 11:57 UTC · model grok-4.3

classification 🧮 math.SP math.PR

keywords Toeplitz matricescompletely monotone sequencesgeneralized eigenvectorsspectral expansioninfinite matricesnon-normal operators

0 comments

The pith

Infinite Toeplitz matrices whose entries come from completely monotone sequences admit a generalized eigenvector expansion that uses only real eigenvalues and real eigenvectors.

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

The paper studies infinite Toeplitz matrices T whose off-diagonals are generated by two completely monotone sequences. It derives explicit formulas for the generalized eigenvectors of T and proves that these formulas, together with the corresponding eigenvalues, produce an expansion of the action of T. The resulting expressions remain real-valued even when T fails to be normal. A reader would care because the result supplies a concrete spectral representation for a broad class of non-self-adjoint operators that arise in discrete convolution problems.

Core claim

When both (a_k) and (a_{-k}) are completely monotone, the infinite Toeplitz matrix T = (a_{k-l}) possesses a family of generalized eigenvectors that can be written with real entries; these vectors, paired with their (real) eigenvalues, furnish a generalized eigenvector expansion for T.

What carries the argument

The generalized eigenvectors constructed directly from the completely monotone sequences, which serve as the building blocks for the expansion of the operator T.

If this is right

The expansion applies uniformly to normal and non-normal Toeplitz matrices under the monotonicity hypothesis.
All eigenvalues appearing in the expansion are real numbers.
All eigenvectors appearing in the expansion have only real components.
The representation reduces to an ordinary eigenvector expansion when T happens to be normal.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same monotonicity condition may allow explicit computation of the resolvent or the semigroup generated by T.
The result could extend to block-Toeplitz matrices or to matrices with entries that are mixtures of completely monotone sequences.
Numerical schemes that truncate the infinite matrix might inherit stability from the real-valued eigenbasis.

Load-bearing premise

The two sequences that define the entries of the Toeplitz matrix must both be completely monotone.

What would settle it

An explicit pair of completely monotone sequences for which the derived real-valued generalized eigenvectors fail to span the action of the corresponding Toeplitz matrix.

Figures

Figures reproduced from arXiv: 2606.01986 by Jacek Wszo{\l}a, Mateusz Kwa\'snicki.

**Figure 1.** Figure 1: The spine Γ and the generalised spectrum Λ for the 𝒜ℳ/𝒞ℳ sequence with generating function 𝑎ˆ(𝑧) = −(1 − 𝑧) 𝛼 (1 − 𝑧 −1 ) 𝛽 , where 𝛼 = 0.4 and 𝛽 = 0.8: (a) the spine Γ (dashed line), its mirror image Γ (dotted line), the unit circle (solid line), and the unit disk (gray region); (b) their image under −𝑎ˆ: the generalised spectrum Λ of −𝑇 (dashed line on the real axis), the essential spectrum of −𝑇 (solid… view at source ↗

**Figure 2.** Figure 2: (a) The spine Γ𝑓 (purple line) of a Rogers function 𝑓 . (b) The spine Γ (solid purple line), the symmetrised spine Γ ★ (solid and dashed purple line), and the regions 𝐷 + 𝐹 (red) and 𝐷 − 𝐹 (blue), for the symbol 𝐹 of an 𝒜ℳ/𝒞ℳ sequence. (c) The spine Γ𝑓 is the union of pairwise disjoint simple real-analytic curves, which begin and end at the imaginary axis or at infinity. Furthermore, Γ𝑓 has parameterisati… view at source ↗

**Figure 3.** Figure 3: (a) The curves parameterised by 𝛾+ (cyan) and 𝛾− (magenta). (b) The curve parameterised by 𝛾− (magenta) and its image under inversion with respect to the unit circle, parameterised by 𝛾ˇ+ (cyan). have Γ = {𝛾+(𝜆) : 𝜆 ∈ Λ}, (4.23) Λ = {𝜆 ∈ (0, 𝐿+) : Im𝛾+(𝜆) > 0}. (4.24) Finally, consider 𝜆 ∈ (0, 𝐿+) \ 𝜕Λ. Denote 𝑧 = 𝛾+(𝜆) and 𝑟 = 𝜆 −1 𝑓 (𝜆). Then 𝑟 ∈ (0, ∞) \ 𝑍𝑓 , and hence, by Proposition 4.8, we have 𝜁𝑓 (… view at source ↗

**Figure 4.** Figure 4: An interval [𝑥1, 𝑥2] in Step 5 of the proof of Proposition 4.12. for 𝜆 ∈ Λ and hence, by continuity, for 𝜆 ∈ Cl Λ. By definition, the above equality also holds for 𝜆 = 0: 𝛾+(0) = 𝛾−(0) = 1. In Steps 5 and 6, we extend (4.27) to 𝜆 ∈ [0, 𝐿] for an appropriate 𝐿. Step 4. Consider 𝜆1, 𝜆2 ∈ {0}∪𝜕Λ such that 𝜆1 < 𝜆2. Since (4.27) holds for 𝜆 = 𝜆1 and 𝜆 = 𝜆2, we may define 𝑥1 = 𝛾+(𝜆1) = 𝛾−(𝜆1) and 𝑥2 = 𝛾+(𝜆2) = 𝛾… view at source ↗

**Figure 5.** Figure 5: The six scenarios in Step 6 in the proof of Proposition 4.12. It follows that for every 𝜆 ∈ (𝜆1, 𝜆2), both 𝑥 = 𝛾+(𝜆) and 𝑥 = 𝛾−(𝜆) satisfy 𝑥 ∈ (𝑥1, 𝑥2) and 𝐹 (𝑥) = 𝜆. Since 𝐹 is strictly monotone on (𝑥1, 𝑥2), we have𝛾+(𝜆) = 𝛾−(𝜆), and so formula (4.27) holds for 𝜆 ∈ (𝜆1, 𝜆2). Since (𝜆1, 𝜆2) is an arbitrary maximal interval in (0,sup Λ) \ Cl Λ, and (4.27) holds for 𝜆 ∈ {0} ∪ Cl Λ, we conclude that in fact (… view at source ↗

read the original abstract

We study the spectral theory of infinite Toeplitz matrices $T = (a_{k - l})$ under the assumption that $(a_k)$ and $(a_{-k})$ are completely monotone sequences. We derive expressions for generalised eigenvectors and prove a generalised eigenvector expansion of $T$. Even if the matrix $T$ is not normal, our expressions involve only eigenvalues and eigenvectors with real entries.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The central claim of real-valued generalized eigenvectors for non-normal Toeplitz operators under separate complete monotonicity on positive and negative indices conflicts with the symbol taking complex values.

read the letter

The colleague should know that this paper asserts explicit expressions for generalized eigenvectors of bilateral infinite Toeplitz matrices that use only real eigenvalues and real eigenvectors, even when T is non-normal, and proves an expansion under the assumption that both (a_k) and (a_{-k}) are completely monotone. The stress-test note correctly flags that this runs into trouble.

The new material is the derivation of those expressions and the expansion theorem. Complete monotonicity supplies Hausdorff integral representations for the sequences, which the authors presumably exploit to write down the eigenvectors in closed form. That is the concrete technical step the abstract highlights.

The paper does a reasonable job of isolating a class where positivity and monotonicity might make spectral expansions tractable. If the derivations are correct on their own terms, the formulas could serve as a calculational tool inside that narrow setting.

The soft spot is load-bearing and sits at the claim itself. When the two sequences are independent, the symbol ϕ(θ) = ∑ a_k e^{-i k θ} is the sum of two distinct geometric series and takes non-real values on the unit circle. The spectrum of T is then the essential range of ϕ and is not real. The abstract nevertheless states that the expressions involve only real entries. No symmetry between a_k and a_{-k} is mentioned, so the real-valued conclusion does not follow from the stated hypotheses. This is not a minor technical slip; it directly undercuts the main selling point.

The work is aimed at a small group already working on spectral questions for Toeplitz operators with monotone coefficients. A specialist might extract a usable formula if the real-spectrum issue is repaired, but the current version does not hold together. I would not bring it to a reading group, would not cite it, and would not send it to peer review without a substantial rewrite that either adds the missing symmetry assumption or explains how real eigenvectors survive a complex spectrum.

Referee Report

1 major / 0 minor

Summary. The paper studies the spectral theory of infinite Toeplitz matrices T = (a_{k-l}) on ℓ²(ℤ) under the assumption that both (a_k) for k ≥ 0 and (a_{-k}) for k ≥ 0 are completely monotone sequences. It derives explicit expressions for generalised eigenvectors and proves a generalised eigenvector expansion for T. The central claim is that these expressions involve only real eigenvalues and real eigenvectors, even when T is non-normal.

Significance. If the central claims were correct, the work would provide a real-valued spectral calculus for a class of non-normal structured operators defined by monotone sequences, which could be useful in contexts where such sequences arise. However, the completely monotone hypothesis does not guarantee a real spectrum when the two sequences are distinct, limiting the potential impact.

major comments (1)

[Abstract] Abstract: The claim that the generalised eigenvector expressions 'involve only eigenvalues and eigenvectors with real entries' even when T is non-normal is contradicted by the spectral theory of bilateral Toeplitz operators. When the Hausdorff measures μ and ν for (a_k) and (a_{-k}) differ, the symbol ϕ(θ) = ∑_{k∈ℤ} a_k e^{-ikθ} takes non-real values on the unit circle, so σ(T) is not contained in ℝ. This directly undermines the stated expansion.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting this important point about the spectral properties. We address the major comment below.

read point-by-point responses

Referee: [Abstract] Abstract: The claim that the generalised eigenvector expressions 'involve only eigenvalues and eigenvectors with real entries' even when T is non-normal is contradicted by the spectral theory of bilateral Toeplitz operators. When the Hausdorff measures μ and ν for (a_k) and (a_{-k}) differ, the symbol ϕ(θ) = ∑_{k∈ℤ} a_k e^{-ikθ} takes non-real values on the unit circle, so σ(T) is not contained in ℝ. This directly undermines the stated expansion.

Authors: We agree that the referee's observation is correct: when the Hausdorff measures μ and ν differ, the symbol ϕ(θ) can take non-real values on the unit circle, so the spectrum of T need not be contained in the reals. This means the claim in the abstract (and echoed in the introduction) that the generalised eigenvector expressions involve only real eigenvalues and eigenvectors is not accurate in general under the paper's hypotheses. We will revise the manuscript to remove this unqualified claim. The revised version will state that the real-valued character of the eigenvalues and eigenvectors holds when the two completely monotone sequences coincide (i.e., when T is self-adjoint), and will note that the explicit expansion construction remains valid more generally, albeit with possibly complex eigenvalues when μ ≠ ν. We will update the abstract, introduction, and any related statements accordingly. revision: yes

Circularity Check

0 steps flagged

No circularity: direct derivation from completely monotone assumption

full rationale

The paper states an explicit assumption that both (a_k) and (a_{-k}) are completely monotone sequences, then derives expressions for generalized eigenvectors and an expansion theorem under that hypothesis. No self-definitional steps, no fitted parameters renamed as predictions, and no load-bearing self-citations are present in the abstract or described claims. The derivation chain is self-contained against the stated assumption and standard integral representations of completely monotone sequences; the result does not reduce to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The sole load-bearing premise visible in the abstract is the complete monotonicity of the two sequences; no free parameters or invented entities are mentioned.

axioms (1)

domain assumption The sequences (a_k) and (a_{-k}) are completely monotone sequences.
This assumption is explicitly required by the abstract for the real-valued generalised eigenvector expansion to hold.

pith-pipeline@v0.9.1-grok · 5586 in / 1112 out tokens · 23997 ms · 2026-06-28T11:57:48.618738+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

25 extracted references · 21 canonical work pages · 1 internal anchor

[1]

Manuel Bogoya, Albrecht Böttcher, Sergei M.\@ Grudsky, Asymptotic Eigenvalue Expansions for Toeplitz Matrices with Certain Fisher--Hartwig Symbols. J. Math.\@ Sci.\@ 271 (2023): 176--196. doi:10.1007/s10958-023-06362-9

work page doi:10.1007/s10958-023-06362-9 2023
[2]

Numer.\@ Linear Algebra Appl.\@ 30(5) (2023) e2496

Manuel Bogoya, Sergei M.\@ Grudsky, Asymptotics for the eigenvalues of Toeplitz matrices with a symbol having a power singularity. Numer.\@ Linear Algebra Appl.\@ 30(5) (2023) e2496. doi:10.1002/nla.2496

work page doi:10.1002/nla.2496 2023
[3]

Sb.\@ Math.\@ 208(11) (2017): 1578--1601; translation from Mat.\@ Sb.\@ 208(11) (2017): 4--28

Albrecht Böttcher, Johan M.\@ Bogoya, Sergei M.\@ Grudsky, Egor A.\@ Maximenko, Asymptotics of eigenvalues and eigenvectors of Toeplitz matrices. Sb.\@ Math.\@ 208(11) (2017): 1578--1601; translation from Mat.\@ Sb.\@ 208(11) (2017): 4--28. doi:10.1070/SM8865

work page doi:10.1070/sm8865 2017
[4]

Springer, New York, 1999

Albrecht Böttcher, Bernd Silberman, Introduction to Large Truncated Toeplitz Matrices. Springer, New York, 1999. doi:10.1007/978-1-4612-1426-7

work page doi:10.1007/978-1-4612-1426-7 1999
[5]

Springer, Berlin, 2006

Albrecht Böttcher, Bernd Silberman, Analysis of Toeplitz Operators. Springer, Berlin, 2006. doi:10.1007/3-540-32436-4

work page doi:10.1007/3-540-32436-4 2006
[6]

Kadanoff, Asymptotics of eigenvalues and eigenvectors of Toeplitz matrices

Hui Dai, Zachary Geary, Leo P. Kadanoff, Asymptotics of eigenvalues and eigenvectors of Toeplitz matrices. J. Stat.\@ Mech.\@ Theory Exp.\@ (2009), P05012: 1--25. doi:10.1088/1742-5468/2009/05/P05012

work page doi:10.1088/1742-5468/2009/05/p05012 2009
[7]

De Carli, S.\,M

L. De Carli, S.\,M. Hudson, Geometric remarks on the level curves of harmonic functions. Bull.\@ London Math.\@ Soc.\@ 42(1) (2010): 83--95. doi:10.1112/blms/bdp099

work page doi:10.1112/blms/bdp099 2010
[8]

Duren, Theory of H^p spaces

Peter L. Duren, Theory of H^p spaces. Academic Press, New York, London, 1970. 978-0122251504

1970
[9]

Gakhov, Boundary Value Problems

F.\,D. Gakhov, Boundary Value Problems. Pergamon Press, Oxford, 1966

1966
[10]

Rev.\@ Math.\@ Phys.\@ 25(8) (2013), 1350014

Kamil Kaleta, Mateusz Kwaśnicki, Jacek Małecki, One-dimensional quasi-relativistic particle in the box. Rev.\@ Math.\@ Phys.\@ 25(8) (2013), 1350014. doi:10.1142/S0129055X13500141

work page doi:10.1142/s0129055x13500141 2013
[11]

Kamil Kaleta, Mateusz Kwaśnicki, Jacek Małecki, Asymptotic estimate of eigenvalues of pseudo-differential operators in an interval. J. Math.\@ Anal.\@ Appl.\@ 439(2) (2016): 896--924. doi:10.1016/j.jmaa.2016.03.010

work page doi:10.1016/j.jmaa.2016.03.010 2016
[12]

Studia Math.\@ 206(3) (2011): 211--271

Mateusz Kwaśnicki, Spectral analysis of subordinate Brownian motions on the half-line. Studia Math.\@ 206(3) (2011): 211--271. doi:10.4064/sm206-3-2

work page doi:10.4064/sm206-3-2 2011
[13]

Mateusz Kwaśnicki, Eigenvalues of the fractional Laplace operator in the interval. J. Funct.\@ Anal.\@ 262(5) (2012): 2379--2402. doi:10.1016/j.jfa.2011.12.004

work page doi:10.1016/j.jfa.2011.12.004 2012
[14]

Electron.\@ J

Mateusz Kwaśnicki, Fluctuation theory for Lévy processes with completely monotone jumps. Electron.\@ J. Probab.\@ 24 (2019), no.\@ 40: 1--40. doi:10.1214/19-EJP300

work page doi:10.1214/19-ejp300 2019
[15]

Mateusz Kwaśnicki, Suprema of Lévy processes with completely monotone jumps: spectral-theoretic approach. J. Spectral Theory, 15(1) (2025): 13--122. doi:10.4171/jst/546

work page doi:10.4171/jst/546 2025
[16]

Stoch.\@ Proc.\@ Appl.\@ 123 (2013): 1820--1850

Mateusz Kwaśnicki, Jacek Małecki, Michał Ryznar, First passage times for subordinate Brownian motions. Stoch.\@ Proc.\@ Appl.\@ 123 (2013): 1820--1850. doi:10.1016/j.spa.2013.01.011

work page doi:10.1016/j.spa.2013.01.011 2013
[17]

Studia Math.\@ 271(2) (2023): 151--185

Mateusz Kwaśnicki, Jacek Wszoła, Bell-shaped sequences. Studia Math.\@ 271(2) (2023): 151--185. doi:10.4064/sm220923-2-2

work page doi:10.4064/sm220923-2-2 2023
[18]

Math.\@ Z.\@ 311 (2025), no.\@ 62: 1--42

Mateusz Kwaśnicki, Jacek Wszoła, Two-sided bell-shaped sequences. Math.\@ Z.\@ 311 (2025), no.\@ 62: 1--42. doi:10.1007/s00209-025-03829-0

work page doi:10.1007/s00209-025-03829-0 2025
[19]

McCoy, Tai Tsun Wu, The Two-Dimensional Ising Model

Barry M. McCoy, Tai Tsun Wu, The Two-Dimensional Ising Model. Harvard University Press, 1973. doi:10.4159/harvard.9780674180758

work page doi:10.4159/harvard.9780674180758 1973
[20]

Amer.\@ J

Marvin Rosenblum, A concrete spectral theory for self-adjoint Toeplitz operators. Amer.\@ J. Math.\@ 87 (1965): 709--718. doi:10.2307/2373070

work page doi:10.2307/2373070 1965
[21]

Sobolev, Dmitri Yafaev, On spectral analysis of self-adjoint Toeplitz operators

Alexander V. Sobolev, Dmitri Yafaev, On spectral analysis of self-adjoint Toeplitz operators. J. Oper.\@ Theory 84(2) (2020): 453--485. doi:10.7900/jot.2019jun19.2244

work page doi:10.7900/jot.2019jun19.2244 2020
[22]

Sobolev, Dmitri Yafaev, Multichannel scattering theory for Toeplitz operators with piecewise continuous symbols

Alexander V. Sobolev, Dmitri Yafaev, Multichannel scattering theory for Toeplitz operators with piecewise continuous symbols. Anal.\@ PDE 15(6) (2022): 1457--1486. doi:10.2140-apde/2022.15.1457

2022
[23]

De Gruyter, 2026

René L.\@ Schilling, Renming Song, Zoran Vondraček, Bernstein Functions: Theory and Applications. De Gruyter, 2026. doi:10.1515/9783111295121

work page doi:10.1515/9783111295121 2026
[24]

On the curvature of level sets of harmonic functions

Stefan Steinerberger, On the curvature of level sets of harmonic functions. Preprint, 2013. arXiv:1307.2069

work page internal anchor Pith review Pith/arXiv arXiv 2013
[25]

Trefethen, Mark Embree, Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators

Lloyd N. Trefethen, Mark Embree, Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators. Princeton University Press, Princeton, NJ, 2005. 9780691119465

2005

[1] [1]

Manuel Bogoya, Albrecht Böttcher, Sergei M.\@ Grudsky, Asymptotic Eigenvalue Expansions for Toeplitz Matrices with Certain Fisher--Hartwig Symbols. J. Math.\@ Sci.\@ 271 (2023): 176--196. doi:10.1007/s10958-023-06362-9

work page doi:10.1007/s10958-023-06362-9 2023

[2] [2]

Numer.\@ Linear Algebra Appl.\@ 30(5) (2023) e2496

Manuel Bogoya, Sergei M.\@ Grudsky, Asymptotics for the eigenvalues of Toeplitz matrices with a symbol having a power singularity. Numer.\@ Linear Algebra Appl.\@ 30(5) (2023) e2496. doi:10.1002/nla.2496

work page doi:10.1002/nla.2496 2023

[3] [3]

Sb.\@ Math.\@ 208(11) (2017): 1578--1601; translation from Mat.\@ Sb.\@ 208(11) (2017): 4--28

Albrecht Böttcher, Johan M.\@ Bogoya, Sergei M.\@ Grudsky, Egor A.\@ Maximenko, Asymptotics of eigenvalues and eigenvectors of Toeplitz matrices. Sb.\@ Math.\@ 208(11) (2017): 1578--1601; translation from Mat.\@ Sb.\@ 208(11) (2017): 4--28. doi:10.1070/SM8865

work page doi:10.1070/sm8865 2017

[4] [4]

Springer, New York, 1999

Albrecht Böttcher, Bernd Silberman, Introduction to Large Truncated Toeplitz Matrices. Springer, New York, 1999. doi:10.1007/978-1-4612-1426-7

work page doi:10.1007/978-1-4612-1426-7 1999

[5] [5]

Springer, Berlin, 2006

Albrecht Böttcher, Bernd Silberman, Analysis of Toeplitz Operators. Springer, Berlin, 2006. doi:10.1007/3-540-32436-4

work page doi:10.1007/3-540-32436-4 2006

[6] [6]

Kadanoff, Asymptotics of eigenvalues and eigenvectors of Toeplitz matrices

Hui Dai, Zachary Geary, Leo P. Kadanoff, Asymptotics of eigenvalues and eigenvectors of Toeplitz matrices. J. Stat.\@ Mech.\@ Theory Exp.\@ (2009), P05012: 1--25. doi:10.1088/1742-5468/2009/05/P05012

work page doi:10.1088/1742-5468/2009/05/p05012 2009

[7] [7]

De Carli, S.\,M

L. De Carli, S.\,M. Hudson, Geometric remarks on the level curves of harmonic functions. Bull.\@ London Math.\@ Soc.\@ 42(1) (2010): 83--95. doi:10.1112/blms/bdp099

work page doi:10.1112/blms/bdp099 2010

[8] [8]

Duren, Theory of H^p spaces

Peter L. Duren, Theory of H^p spaces. Academic Press, New York, London, 1970. 978-0122251504

1970

[9] [9]

Gakhov, Boundary Value Problems

F.\,D. Gakhov, Boundary Value Problems. Pergamon Press, Oxford, 1966

1966

[10] [10]

Rev.\@ Math.\@ Phys.\@ 25(8) (2013), 1350014

Kamil Kaleta, Mateusz Kwaśnicki, Jacek Małecki, One-dimensional quasi-relativistic particle in the box. Rev.\@ Math.\@ Phys.\@ 25(8) (2013), 1350014. doi:10.1142/S0129055X13500141

work page doi:10.1142/s0129055x13500141 2013

[11] [11]

Kamil Kaleta, Mateusz Kwaśnicki, Jacek Małecki, Asymptotic estimate of eigenvalues of pseudo-differential operators in an interval. J. Math.\@ Anal.\@ Appl.\@ 439(2) (2016): 896--924. doi:10.1016/j.jmaa.2016.03.010

work page doi:10.1016/j.jmaa.2016.03.010 2016

[12] [12]

Studia Math.\@ 206(3) (2011): 211--271

Mateusz Kwaśnicki, Spectral analysis of subordinate Brownian motions on the half-line. Studia Math.\@ 206(3) (2011): 211--271. doi:10.4064/sm206-3-2

work page doi:10.4064/sm206-3-2 2011

[13] [13]

Mateusz Kwaśnicki, Eigenvalues of the fractional Laplace operator in the interval. J. Funct.\@ Anal.\@ 262(5) (2012): 2379--2402. doi:10.1016/j.jfa.2011.12.004

work page doi:10.1016/j.jfa.2011.12.004 2012

[14] [14]

Electron.\@ J

Mateusz Kwaśnicki, Fluctuation theory for Lévy processes with completely monotone jumps. Electron.\@ J. Probab.\@ 24 (2019), no.\@ 40: 1--40. doi:10.1214/19-EJP300

work page doi:10.1214/19-ejp300 2019

[15] [15]

Mateusz Kwaśnicki, Suprema of Lévy processes with completely monotone jumps: spectral-theoretic approach. J. Spectral Theory, 15(1) (2025): 13--122. doi:10.4171/jst/546

work page doi:10.4171/jst/546 2025

[16] [16]

Stoch.\@ Proc.\@ Appl.\@ 123 (2013): 1820--1850

Mateusz Kwaśnicki, Jacek Małecki, Michał Ryznar, First passage times for subordinate Brownian motions. Stoch.\@ Proc.\@ Appl.\@ 123 (2013): 1820--1850. doi:10.1016/j.spa.2013.01.011

work page doi:10.1016/j.spa.2013.01.011 2013

[17] [17]

Studia Math.\@ 271(2) (2023): 151--185

Mateusz Kwaśnicki, Jacek Wszoła, Bell-shaped sequences. Studia Math.\@ 271(2) (2023): 151--185. doi:10.4064/sm220923-2-2

work page doi:10.4064/sm220923-2-2 2023

[18] [18]

Math.\@ Z.\@ 311 (2025), no.\@ 62: 1--42

Mateusz Kwaśnicki, Jacek Wszoła, Two-sided bell-shaped sequences. Math.\@ Z.\@ 311 (2025), no.\@ 62: 1--42. doi:10.1007/s00209-025-03829-0

work page doi:10.1007/s00209-025-03829-0 2025

[19] [19]

McCoy, Tai Tsun Wu, The Two-Dimensional Ising Model

Barry M. McCoy, Tai Tsun Wu, The Two-Dimensional Ising Model. Harvard University Press, 1973. doi:10.4159/harvard.9780674180758

work page doi:10.4159/harvard.9780674180758 1973

[20] [20]

Amer.\@ J

Marvin Rosenblum, A concrete spectral theory for self-adjoint Toeplitz operators. Amer.\@ J. Math.\@ 87 (1965): 709--718. doi:10.2307/2373070

work page doi:10.2307/2373070 1965

[21] [21]

Sobolev, Dmitri Yafaev, On spectral analysis of self-adjoint Toeplitz operators

Alexander V. Sobolev, Dmitri Yafaev, On spectral analysis of self-adjoint Toeplitz operators. J. Oper.\@ Theory 84(2) (2020): 453--485. doi:10.7900/jot.2019jun19.2244

work page doi:10.7900/jot.2019jun19.2244 2020

[22] [22]

Sobolev, Dmitri Yafaev, Multichannel scattering theory for Toeplitz operators with piecewise continuous symbols

Alexander V. Sobolev, Dmitri Yafaev, Multichannel scattering theory for Toeplitz operators with piecewise continuous symbols. Anal.\@ PDE 15(6) (2022): 1457--1486. doi:10.2140-apde/2022.15.1457

2022

[23] [23]

De Gruyter, 2026

René L.\@ Schilling, Renming Song, Zoran Vondraček, Bernstein Functions: Theory and Applications. De Gruyter, 2026. doi:10.1515/9783111295121

work page doi:10.1515/9783111295121 2026

[24] [24]

On the curvature of level sets of harmonic functions

Stefan Steinerberger, On the curvature of level sets of harmonic functions. Preprint, 2013. arXiv:1307.2069

work page internal anchor Pith review Pith/arXiv arXiv 2013

[25] [25]

Trefethen, Mark Embree, Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators

Lloyd N. Trefethen, Mark Embree, Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators. Princeton University Press, Princeton, NJ, 2005. 9780691119465

2005