Trace formulas in higher dimensions

Alexandr Usachev; Arup Chattopadhyay; Chandan Pradhan; Saikat Giri

arxiv: 2303.13298 · v6 · pith:5LEGCVKKnew · submitted 2023-03-23 · 🧮 math.FA

Trace formulas in higher dimensions

Arup Chattopadhyay , Saikat Giri , Chandan Pradhan , Alexandr Usachev This is my paper

Pith reviewed 2026-05-24 10:02 UTC · model grok-4.3

classification 🧮 math.FA

keywords trace formulasKrein formulaKoplienko formulamultivariable operator functionssymmetrically normed idealsself-adjoint operatorsmaximal dissipative operatorssingular traces

0 comments

The pith

Krein and Koplienko trace formulas extend to multivariable operator functions on symmetrically normed ideals for self-adjoint and maximal dissipative operators.

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

The paper proves that the classical one-variable trace formulas of Krein and Koplienko admit direct extensions to functions of several variables when the operators lie in symmetrically normed ideals. The extensions hold for both normal traces and singular traces, and the scalar functions may be either analytic or non-analytic. A reader would care because these identities supply explicit integral expressions for the trace of a perturbed multivariable function, allowing concrete calculations in spectral theory and perturbation problems that involve multiple operators at once.

Core claim

The central claim is that the Krein trace formula and the Koplienko trace formula both possess multivariable analogues for operator functions acting on symmetrically normed ideals of bounded operators; the identities are established for self-adjoint operators and for maximal dissipative operators, they apply equally to ideals equipped with normal traces and to those equipped with singular traces, and the admissible scalar functions include both analytic and non-analytic classes, with the results illustrated by examples.

What carries the argument

The multivariable Krein and Koplienko trace formulas, which equate the trace of a function of several operators to an integral involving the joint perturbation or spectral measure of those operators.

If this is right

Trace computations become available for non-commuting as well as commuting tuples of operators inside the stated ideals.
The formulas apply equally when the trace on the ideal is singular rather than normal.
Both analytic and non-analytic scalar functions of several variables are covered by the same identities.
Explicit examples confirm that the multivariable formulas recover known one-variable cases as special instances.

Where Pith is reading between the lines

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

The same integral expressions could be used to bound traces arising in finite-dimensional matrix models that approximate infinite-dimensional operators.
Numerical verification on low-dimensional matrices with chosen self-adjoint pairs would provide a direct check of the multivariable extension.
The results suggest that similar trace identities might exist for other classes of operators once suitable admissible function spaces are identified.

Load-bearing premise

The scalar functions must belong to admissible classes that allow both analytic and non-analytic cases while the operators remain self-adjoint or maximal dissipative.

What would settle it

A concrete pair of self-adjoint operators in a symmetrically normed ideal together with an admissible multivariable function for which the difference of traces fails to equal the integral expression given by the claimed formula.

read the original abstract

The paper establishes the Krein and Koplienko trace formulas for multivariable operator functions on symmetrically normed ideals of bounded operators. Results are proved for self-adjoint and maximal dissipative operators. They cover both ideals with normal and singular traces. The admissible function classes considered in the trace formulas include both analytic and non-analytic scalar functions. Results are illustrated with examples.

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 paper extends Krein and Koplienko trace formulas to multivariable operator functions on symmetrically normed ideals, with no clear internal problems from the description.

read the letter

The main point is that this work establishes multivariable versions of the Krein and Koplienko trace formulas for operator functions. It handles self-adjoint and maximal dissipative operators, covers both normal and singular traces, and includes both analytic and non-analytic scalar functions from admissible classes, with some examples at the end. The extension to several variables is the actual new element, since the one-variable cases are already in the literature. The setup stays within symmetrically normed ideals, which keeps the claims grounded. The stress-test note found no inconsistencies or hidden steps, and the abstract aligns with standard directions in perturbation theory for operators. The assumptions on the function classes and operator restrictions are stated up front without signs of circularity. The soft spot is that the abstract gives no proof details or explicit technical conditions, so the real strength depends on whether the full arguments close the gaps that often appear when moving from one variable to several. If the proofs are complete and the examples check out, this is honest incremental progress rather than a loose claim. This is for people already working on trace formulas and operator ideals in functional analysis. A reader who follows the one-variable results would get direct value from seeing how the multivariable case is handled. It deserves a serious referee because the claims are concrete and the setting is standard, even though the scope is limited.

Referee Report

1 major / 0 minor

Summary. The manuscript establishes the Krein and Koplienko trace formulas for multivariable operator functions on symmetrically normed ideals of bounded operators. Results are proved for self-adjoint and maximal dissipative operators. They cover both ideals with normal and singular traces. The admissible function classes considered in the trace formulas include both analytic and non-analytic scalar functions. Results are illustrated with examples.

Significance. If the derivations hold, the work extends classical single-variable trace formulas to the multivariable setting on symmetrically normed ideals, addressing both normal and singular traces as well as analytic and non-analytic admissible functions. This could strengthen tools in perturbation theory for operator ideals. The explicit coverage of maximal dissipative operators alongside self-adjoint ones broadens applicability. No machine-checked proofs or parameter-free derivations are mentioned.

major comments (1)

[Abstract] The provided abstract states the claims but contains no derivations, technical conditions, or proof sketches. Without these, it is impossible to confirm that the multivariable extensions are correctly derived or that the admissible classes are properly handled for both normal and singular traces (§1 and main theorems). This directly affects the central claim.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their review. We address the single major comment below.

read point-by-point responses

Referee: [Abstract] The provided abstract states the claims but contains no derivations, technical conditions, or proof sketches. Without these, it is impossible to confirm that the multivariable extensions are correctly derived or that the admissible classes are properly handled for both normal and singular traces (§1 and main theorems). This directly affects the central claim.

Authors: The abstract is a concise summary of the paper's main results, as is conventional. The full derivations, technical conditions on the admissible function classes, and proof sketches for the multivariable Krein and Koplienko formulas (covering self-adjoint and maximal dissipative operators, symmetrically normed ideals, and both normal and singular traces) are provided in §1 and the statements/proofs of the main theorems in the body of the manuscript. revision: no

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper establishes Krein and Koplienko trace formulas for multivariable operator functions on symmetrically normed ideals, for self-adjoint and maximal dissipative operators, covering normal/singular traces and analytic/non-analytic admissible functions. No load-bearing steps reduce by definition, fitted inputs renamed as predictions, or self-citation chains that replace independent derivation. The claimed results align with standard directions in perturbation theory for operator ideals without the abstract or described scope indicating any reduction of outputs to inputs by construction. This is the expected honest non-finding for a paper whose central claims rest on external mathematical structure rather than internal redefinition.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities can be identified because only the abstract is available.

pith-pipeline@v0.9.0 · 5576 in / 982 out tokens · 28503 ms · 2026-05-24T10:02:30.012208+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

38 extracted references · 38 canonical work pages

[1]

V. M. Adamjan and B. S. Pavlov, Trace formula for dissipative operators , Vestnik Leningrad. Univ. Mat. Mekh. Astronom., (1979), no. 2, 5–9, 118

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[2]

A. B. Aleksandrov, V. V. Peller and D. S. Potapov, On a trace formula for functions of noncommuting operators, Mat. Zametki, 106 (2019), no. 4, 483–490 (in Russian); English transl. in Math. Notes , 106 (2019), no. 3-4, 481–487

work page 2019
[3]

N. A. Azamov, A. L. Carey, P. G. Dodds and F. A. Sukochev, Operator integrals, spectral shift, and spectral ﬂow, Canad. J. Math., 61 (2009), no. 2, 241–263

work page 2009
[4]

N. A. Azamov, A. L. Carey and F. A. Sukochev, The spectral shift function and spectral ﬂow , Comm. Math. Phys., 276 (2007), no. 1, 51–91

work page 2007
[5]

N. A. Azamov, P. G. Dodds and F. A. Sukochev, The Krein spectral shift function in semiﬁnite von Neumann algebras, Integral Equations Operator Theory, 55 (2006), no. 3, 347–362

work page 2006
[6]

M. S. Birman and M. G. Krein, On the theory of wave operators and scattering operators , Dokl. Akad. Nauk SSSR, 144 (1962), 475–478

work page 1962
[7]

M. S. Birman and M. Z. Solomjak, Spectral theory of self-adjoint operators in Hilbert space , In: Mathe- matics and its Applications (Soviet Series), D. Reidel Publishing Co., Do rdrecht, 1987

work page 1987
[8]

A. L. Carey and F. A. Sukochev, Dixmier traces and some applications to noncommutative geo metry, Uspekhi Mat. Nauk, 61 (2006), no. 6(372), 45–110 (in Russian); English transl. in Russian Math. Surveys, 61 (2006), no. 6, 1039–1099

work page 2006
[9]

Chattopadhyay, S

A. Chattopadhyay, S. Das and C. Pradhan, Second-order trace formulas, Math. Nachr., 297 (2024), no. 7, 2581–2608

work page 2024
[10]

Chattopadhyay, S

A. Chattopadhyay, S. Giri and C. Pradhan, Estimates and higher-order spectral shift measures in seve ral variables, Linear Algebra Appl., 698 (2024), 102–134

work page 2024
[11]

Clark and F

S. Clark and F. Gesztesy, Weyl-Titchmarsh M -function asymptotics, local uniqueness results, trace fo rmu- las, and Borg-type theorems for Dirac operators , Trans. Amer. Math. Soc., 354 (2002), no. 9, 3475–3534

work page 2002
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Connes, Noncommutative geometry, Academic Press, Inc., San Diego, CA, 1994, xiv+661 pp

A. Connes, Noncommutative geometry, Academic Press, Inc., San Diego, CA, 1994, xiv+661 pp

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[13]

Dixmier, Existence de traces non normales , C

J. Dixmier, Existence de traces non normales , C. R. Acad. Sci. Paris S´ er. A-B,262 (1966), A1107–A1108

work page 1966
[14]

P. G. Dodds, B. de Pagter, E. M. Semenov and F. A. Sukochev, Symmetric functionals and singular traces, Positivity, 2 (1998), no. 1, 47–75

work page 1998
[15]

Dykema and A

K. Dykema and A. Skripka, Higher order spectral shift , J. Funct. Anal., 257 (2009), no. 4, 1092–1132

work page 2009
[16]

Dykema and A

K. Dykema and A. Skripka, Perturbation formulas for traces on normed ideals , Comm. Math. Phys., 325 (2014), no. 3, 1107–1138

work page 2014
[17]

Fricain and J

E. Fricain and J. Mashreghi, The theory of H(b) spaces. Vol. 1 , vol. 20, Cambridge University Press, Cambridge, 2016, xix+681 pp

work page 2016
[18]

Gesztesy and B

F. Gesztesy and B. Simon, The xi function , Acta Math., 176 (1996), no. 1, 49–71

work page 1996
[19]

Gesztesy and B

F. Gesztesy and B. Simon, Uniqueness theorems in inverse spectral theory for one-dim ensional Schr¨ odinger operators, Trans. Amer. Math. Soc., 348 (1996), no. 1, 349–373

work page 1996
[20]

Gesztesy and M

F. Gesztesy and M. Zinchenko, A Borg-type theorem associated with orthogonal polynomial s on the unit circle, J. London Math. Soc. (2), 74 (2006), no. 3, 757–777. 36 A. CHATTOPADHYAY, S. GIRI, C. PRADHAN, AND A. USACHEV

work page 2006
[21]

I. C. Gohberg and M. G. Krein, Introduction to the theory of linear nonselfadjoint operat ors. Translations of Mathematical Monographs, Vol. 18., American Mathematical Soc iety, Providence, RI, 1969, xv+378 pp

work page 1969
[22]

L. S. Koplienko, The trace formula for perturbations of nonnuclear type , Sibirsk. Mat. Zh., 25 (1984), no. 5, 62–71 (in Russian)

work page 1984
[23]

M. G. Krein, On the trace formula in perturbation theory , Mat. Sbornik N.S., 33/75 (1953), 597–626 (in Russian)

work page 1953
[24]

M. G. Krein, On perturbation determinants and a trace formula for unitar y and self-adjoint operators , Dokl. Akad. Nauk SSSR, 144 (1962), 268–271 (in Russian)

work page 1962
[25]

I. M. Lifshitz, On a problem of the theory of perturbations connected with qu antum statistics , Uspehi Matem. Nauk (N.S.), 7 (1952), no. 1(47), 171–180 (in Russian)

work page 1952
[26]

S. Lord, F. A. Sukochev and D. Zanin, Advances in Dixmier traces and applications , Advances in Non- commutative Geometry—on the occasion of Alain Connes’ 70th birth day, Springer, Cham, 2019, 491–583

work page 2019
[27]

S. Lord, F. A. Sukochev and D. Zanin, Singular traces. Theory and applications , De Gruyter Studies in Mathematics, vol. 46, De Gruyter, Berlin, 2013, xvi+452 pp

work page 2013
[28]

S. Lord, F. A. Sukochev and D. Zanin, Singular traces. Vol. 1. Theory , De Gruyter Studies in Mathe- matics, vol. 46/1, De Gruyter, Berlin, 2021, xxix+383 pp

work page 2021
[29]

A. B. Patel, A joint spectral theorem for unbounded normal operators , J. Austral. Math. Soc. Ser. A, 34 (1983), no. 2, 203–213

work page 1983
[30]

D. S. Potapov, F. A. Sukochev, A. Tomskova and D. Zanin, Fr´ echet diﬀerentiability of the norm of Lp- spaces associated with arbitrary von Neumann algebras , C. R. Math. Acad. Sci. Paris, 352 (2014), no. 11, 923–927

work page 2014
[31]

D. S. Potapov, F. A. Sukochev, A. Usachev and D. Zanin, Singular traces and perturbation formulae of higher order, J. Funct. Anal., 269 (2015), no. 5, 1441–1481

work page 2015
[32]

Reed and B

M. Reed and B. Simon, Methods of modern mathematical physics. I. Functional anal ysis, Academic Press, New York-London, 1972, xvii+325 pp

work page 1972
[33]

Skripka, Trace formulas for multivariate operator functions , Integral Equations Operator Theory, 81 (2015), no

A. Skripka, Trace formulas for multivariate operator functions , Integral Equations Operator Theory, 81 (2015), no. 4, 559–580

work page 2015
[34]

Skripka and A

A. Skripka and A. Tomskova, Multilinear operator integrals. Theory and applications , Lecture Notes in Mathematics, vol. 2250, Springer, Cham, 2019, xi+190 pp

work page 2019
[35]

F. A. Sukochev and A. Usachev, Dixmier traces and non-commutative analysis , J. Geom. Phys., 105 (2016), 102–122

work page 2016
[36]

Teschl, Trace formulas and inverse spectral theory for Jacobi opera tors, Comm

G. Teschl, Trace formulas and inverse spectral theory for Jacobi opera tors, Comm. Math. Phys., 196 (1998), no. 1, 175–202

work page 1998
[37]

Xia, Principal measures and spectral shift measures associated with Dixmier traces , J

J. Xia, Principal measures and spectral shift measures associated with Dixmier traces , J. Funct. Anal., 283 (2022), no. 10, Paper No. 109667, 38 pp

work page 2022
[38]

Zinchenko, Trace formulas and a Borg-type theorem for CMV operators wit h matrix-valued coeﬃcients, Math

M. Zinchenko, Trace formulas and a Borg-type theorem for CMV operators wit h matrix-valued coeﬃcients, Math. Nachr., 283 (2010), no. 2, 312–329. Department of Mathematics, Indian Institute of Technology Guwahati, Guwahati, 781039, India Email address, A. Chattopadhyay: arupchatt@iitg.ac.in, 2003arupchattopadhyay@gmail.co m Email address, S. Giri: saikat.g...

work page 2010

[1] [1]

V. M. Adamjan and B. S. Pavlov, Trace formula for dissipative operators , Vestnik Leningrad. Univ. Mat. Mekh. Astronom., (1979), no. 2, 5–9, 118

work page 1979

[2] [2]

A. B. Aleksandrov, V. V. Peller and D. S. Potapov, On a trace formula for functions of noncommuting operators, Mat. Zametki, 106 (2019), no. 4, 483–490 (in Russian); English transl. in Math. Notes , 106 (2019), no. 3-4, 481–487

work page 2019

[3] [3]

N. A. Azamov, A. L. Carey, P. G. Dodds and F. A. Sukochev, Operator integrals, spectral shift, and spectral ﬂow, Canad. J. Math., 61 (2009), no. 2, 241–263

work page 2009

[4] [4]

N. A. Azamov, A. L. Carey and F. A. Sukochev, The spectral shift function and spectral ﬂow , Comm. Math. Phys., 276 (2007), no. 1, 51–91

work page 2007

[5] [5]

N. A. Azamov, P. G. Dodds and F. A. Sukochev, The Krein spectral shift function in semiﬁnite von Neumann algebras, Integral Equations Operator Theory, 55 (2006), no. 3, 347–362

work page 2006

[6] [6]

M. S. Birman and M. G. Krein, On the theory of wave operators and scattering operators , Dokl. Akad. Nauk SSSR, 144 (1962), 475–478

work page 1962

[7] [7]

M. S. Birman and M. Z. Solomjak, Spectral theory of self-adjoint operators in Hilbert space , In: Mathe- matics and its Applications (Soviet Series), D. Reidel Publishing Co., Do rdrecht, 1987

work page 1987

[8] [8]

A. L. Carey and F. A. Sukochev, Dixmier traces and some applications to noncommutative geo metry, Uspekhi Mat. Nauk, 61 (2006), no. 6(372), 45–110 (in Russian); English transl. in Russian Math. Surveys, 61 (2006), no. 6, 1039–1099

work page 2006

[9] [9]

Chattopadhyay, S

A. Chattopadhyay, S. Das and C. Pradhan, Second-order trace formulas, Math. Nachr., 297 (2024), no. 7, 2581–2608

work page 2024

[10] [10]

Chattopadhyay, S

A. Chattopadhyay, S. Giri and C. Pradhan, Estimates and higher-order spectral shift measures in seve ral variables, Linear Algebra Appl., 698 (2024), 102–134

work page 2024

[11] [11]

Clark and F

S. Clark and F. Gesztesy, Weyl-Titchmarsh M -function asymptotics, local uniqueness results, trace fo rmu- las, and Borg-type theorems for Dirac operators , Trans. Amer. Math. Soc., 354 (2002), no. 9, 3475–3534

work page 2002

[12] [12]

Connes, Noncommutative geometry, Academic Press, Inc., San Diego, CA, 1994, xiv+661 pp

A. Connes, Noncommutative geometry, Academic Press, Inc., San Diego, CA, 1994, xiv+661 pp

work page 1994

[13] [13]

Dixmier, Existence de traces non normales , C

J. Dixmier, Existence de traces non normales , C. R. Acad. Sci. Paris S´ er. A-B,262 (1966), A1107–A1108

work page 1966

[14] [14]

P. G. Dodds, B. de Pagter, E. M. Semenov and F. A. Sukochev, Symmetric functionals and singular traces, Positivity, 2 (1998), no. 1, 47–75

work page 1998

[15] [15]

Dykema and A

K. Dykema and A. Skripka, Higher order spectral shift , J. Funct. Anal., 257 (2009), no. 4, 1092–1132

work page 2009

[16] [16]

Dykema and A

K. Dykema and A. Skripka, Perturbation formulas for traces on normed ideals , Comm. Math. Phys., 325 (2014), no. 3, 1107–1138

work page 2014

[17] [17]

Fricain and J

E. Fricain and J. Mashreghi, The theory of H(b) spaces. Vol. 1 , vol. 20, Cambridge University Press, Cambridge, 2016, xix+681 pp

work page 2016

[18] [18]

Gesztesy and B

F. Gesztesy and B. Simon, The xi function , Acta Math., 176 (1996), no. 1, 49–71

work page 1996

[19] [19]

Gesztesy and B

F. Gesztesy and B. Simon, Uniqueness theorems in inverse spectral theory for one-dim ensional Schr¨ odinger operators, Trans. Amer. Math. Soc., 348 (1996), no. 1, 349–373

work page 1996

[20] [20]

Gesztesy and M

F. Gesztesy and M. Zinchenko, A Borg-type theorem associated with orthogonal polynomial s on the unit circle, J. London Math. Soc. (2), 74 (2006), no. 3, 757–777. 36 A. CHATTOPADHYAY, S. GIRI, C. PRADHAN, AND A. USACHEV

work page 2006

[21] [21]

I. C. Gohberg and M. G. Krein, Introduction to the theory of linear nonselfadjoint operat ors. Translations of Mathematical Monographs, Vol. 18., American Mathematical Soc iety, Providence, RI, 1969, xv+378 pp

work page 1969

[22] [22]

L. S. Koplienko, The trace formula for perturbations of nonnuclear type , Sibirsk. Mat. Zh., 25 (1984), no. 5, 62–71 (in Russian)

work page 1984

[23] [23]

M. G. Krein, On the trace formula in perturbation theory , Mat. Sbornik N.S., 33/75 (1953), 597–626 (in Russian)

work page 1953

[24] [24]

M. G. Krein, On perturbation determinants and a trace formula for unitar y and self-adjoint operators , Dokl. Akad. Nauk SSSR, 144 (1962), 268–271 (in Russian)

work page 1962

[25] [25]

I. M. Lifshitz, On a problem of the theory of perturbations connected with qu antum statistics , Uspehi Matem. Nauk (N.S.), 7 (1952), no. 1(47), 171–180 (in Russian)

work page 1952

[26] [26]

S. Lord, F. A. Sukochev and D. Zanin, Advances in Dixmier traces and applications , Advances in Non- commutative Geometry—on the occasion of Alain Connes’ 70th birth day, Springer, Cham, 2019, 491–583

work page 2019

[27] [27]

S. Lord, F. A. Sukochev and D. Zanin, Singular traces. Theory and applications , De Gruyter Studies in Mathematics, vol. 46, De Gruyter, Berlin, 2013, xvi+452 pp

work page 2013

[28] [28]

S. Lord, F. A. Sukochev and D. Zanin, Singular traces. Vol. 1. Theory , De Gruyter Studies in Mathe- matics, vol. 46/1, De Gruyter, Berlin, 2021, xxix+383 pp

work page 2021

[29] [29]

A. B. Patel, A joint spectral theorem for unbounded normal operators , J. Austral. Math. Soc. Ser. A, 34 (1983), no. 2, 203–213

work page 1983

[30] [30]

D. S. Potapov, F. A. Sukochev, A. Tomskova and D. Zanin, Fr´ echet diﬀerentiability of the norm of Lp- spaces associated with arbitrary von Neumann algebras , C. R. Math. Acad. Sci. Paris, 352 (2014), no. 11, 923–927

work page 2014

[31] [31]

D. S. Potapov, F. A. Sukochev, A. Usachev and D. Zanin, Singular traces and perturbation formulae of higher order, J. Funct. Anal., 269 (2015), no. 5, 1441–1481

work page 2015

[32] [32]

Reed and B

M. Reed and B. Simon, Methods of modern mathematical physics. I. Functional anal ysis, Academic Press, New York-London, 1972, xvii+325 pp

work page 1972

[33] [33]

Skripka, Trace formulas for multivariate operator functions , Integral Equations Operator Theory, 81 (2015), no

A. Skripka, Trace formulas for multivariate operator functions , Integral Equations Operator Theory, 81 (2015), no. 4, 559–580

work page 2015

[34] [34]

Skripka and A

A. Skripka and A. Tomskova, Multilinear operator integrals. Theory and applications , Lecture Notes in Mathematics, vol. 2250, Springer, Cham, 2019, xi+190 pp

work page 2019

[35] [35]

F. A. Sukochev and A. Usachev, Dixmier traces and non-commutative analysis , J. Geom. Phys., 105 (2016), 102–122

work page 2016

[36] [36]

Teschl, Trace formulas and inverse spectral theory for Jacobi opera tors, Comm

G. Teschl, Trace formulas and inverse spectral theory for Jacobi opera tors, Comm. Math. Phys., 196 (1998), no. 1, 175–202

work page 1998

[37] [37]

Xia, Principal measures and spectral shift measures associated with Dixmier traces , J

J. Xia, Principal measures and spectral shift measures associated with Dixmier traces , J. Funct. Anal., 283 (2022), no. 10, Paper No. 109667, 38 pp

work page 2022

[38] [38]

Zinchenko, Trace formulas and a Borg-type theorem for CMV operators wit h matrix-valued coeﬃcients, Math

M. Zinchenko, Trace formulas and a Borg-type theorem for CMV operators wit h matrix-valued coeﬃcients, Math. Nachr., 283 (2010), no. 2, 312–329. Department of Mathematics, Indian Institute of Technology Guwahati, Guwahati, 781039, India Email address, A. Chattopadhyay: arupchatt@iitg.ac.in, 2003arupchattopadhyay@gmail.co m Email address, S. Giri: saikat.g...

work page 2010