Hostname: page-component-7dd5485656-bt4hw Total loading time: 0 Render date: 2025-10-26T21:30:12.908Z Has data issue: false hasContentIssue false
  • English
  • Français

Higher-order trace formulas for contractive and dissipative operators

Part of: General theory of linear operators

Published online by Cambridge University Press:  15 August 2025

Arup Chattopadhyay ,
Chandan Pradhan Open the ORCID record for Chandan Pradhan [Opens in a new window]  and
Anna Skripka Open the ORCID record for Anna Skripka [Opens in a new window]
Arup Chattopadhyay
Affiliation:
Department of Mathematics, Indian Institute of Technology , Guwahati 781039, India e-mail: arupchatt@iitg.ac.in, 2003arupchattopadhyay@gmail.com
Chandan Pradhan
Affiliation:
Department of Mathematics, Indian Institute of Science , Bengaluru 560012, India e-mail: chandan.pradhan2108@gmail.com, chandanp@iisc.ac.in
Anna Skripka*
Affiliation:
Department of Mathematics and Statistics, University of New Mexico , Albuquerque, NM 87106, United States
*
e-mail: skripka@math.unm.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

We establish higher-order trace formulas for pairs of contractions along a multiplicative path generated by a self-adjoint operator in a Schatten-von Neumann ideal, removing earlier stringent restrictions on the kernel and defect operator of the contractions and significantly enlarging the set of admissible functions. We also derive higher-order trace formulas for maximal dissipative operators under relaxed assumptions and new simplified trace formulas for unitary and resolvent comparable self-adjoint operators. The respective spectral shift measures are absolutely continuous and, in the case of contractions, the set of admissible functions for the nth-order trace formula on the unit circle includes the Besov class $B^n_{\infty , 1}(\mathbb {T})$. Both aforementioned properties are new in the mentioned generality.

Keywords

Spectral shift functionhigher-order trace formulaSchatten-von Neumann perturbationcontractive and dissipative operatormultilinear operator integral

MSC classification

Primary: 47A55: Perturbation theory 47A56: Functions whose values are linear operators (operator and matrix valued functions, etc., including analytic and meromorphic ones)

Information

Type
Article
Information
Canadian Journal of Mathematics , First View , pp. 1 - 26
Check for updates
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

Footnotes

A. Chattopadhyay is supported by the Core Research Grant (CRG), File No: CRG/2023/004826, of SERB. A. Skripka is supported in part by Simons Foundation Grant MP-TSM-00002648. C. Pradhan acknowledges support from the IoE post-doctoral fellowship at IISc Bangalore, as well as the NBHM post-doctoral fellowship (File No. 0204/27/(9)/2023/R&D-II/11882).

References

Adamjan, V. M. and Pavlov, B. S., Trace formula for dissipative operators . Vestn. Leningr. Univ. Math. Mekh. Astronom. 1979(1979), no. 2, 5–9, 118 pp.Google Scholar
Azamov, N. A., Carey, A. L., Dodds, P. G., and Sukochev, F., Operator integrals, spectral shift, and spectral flow . Canad. J. Math. 61(2009), no. 2, 241–263.10.4153/CJM-2009-012-0CrossRefGoogle Scholar
Birman, M. S. and Pushnitski, A. B., Spectral shift function, amazing and multifaceted. Dedicated to the memory of Mark Grigorievich Krein (1907–1989) . Integr. Equ. Oper. Theory 30(1998), no. 2, 191–199.10.1007/BF01238218CrossRefGoogle Scholar
Chattopadhyay, A., Das, S., and Pradhan, C., Second-order trace formulas . Math. Nachr. 297(2024), no. 7, 2581–2608.10.1002/mana.202200295CrossRefGoogle Scholar
Chattopadhyay, A., Coine, C., Giri, S., and Pradhan, C., Higher order ${S}^p$ -differentiability: The unitary case . J. Spectr. Theory 15(2025), no. 1, 195–222.10.4171/jst/536CrossRefGoogle Scholar
Chattopadhyay, A., van Nuland, T. D. H., and Pradhan, C., Differentiation, Taylor series, and all order spectral shift functions, for relatively bounded perturbations. Preprint, 2024. https://arxiv.org/abs/2404.18422.Google Scholar
Chattopadhyay, A. and Pradhan, C., Higher-order spectral shift for pairs of contractions via multiplicative path . New York J. Math. 29(2023), 261–285.Google Scholar
Chattopadhyay, A. and Sinha, K. B., On the Carey-Helton-Howe-Pincus trace formula . J. Funct. Anal. 274(2018), no. 8, 2265–2290.CrossRefGoogle Scholar
Chattopadhyay, A. and Sinha, K. B., Trace formula for contractions and its representation in $D$ . J. Oper. Theory 88(2022), no. 2, 275–288.10.7900/jot.2021may13.2323CrossRefGoogle Scholar
Coine, C., Functions of unitaries with $\mathcal{S}^p$ -perturbations for non continuously differentiable functions, Studia Math. 2024. https://arxiv.org/abs/2406.13333.Google Scholar
Coine, C., Le Merdy, C., and Sukochev, F., When do triple operator integrals take value in the trace class? Ann. Inst. Fourier, Grenoble 71(2021), no. 4, 1393–1448.CrossRefGoogle Scholar
Dykema, K. and Skripka, A., Perturbation formulas for traces on normed ideals . Commun. Math. Phys. 325(2014), no. 3, 1107–1138.10.1007/s00220-013-1828-5CrossRefGoogle Scholar
de Pagter, B. and Sukochev, F. A., Differentiation of operator functions in non-commutative ${L}_p$ -spaces . J. Funct. Anal. 212(2004), no. 1, 28–75.CrossRefGoogle Scholar
Gesztesy, F., Pushnitski, A., and Simon, B., On the Koplienko spectral shift function I. Basics . Zh. Mat. Fiz. Anal. Geom. 4(2008), no. 1, 63–107, 202.Google Scholar
Gohberg, I. C. and Krein, M. G., Introduction to the theory of linear nonselfadjoint operators . In: Translations of mathematical monographs. vol. 18. American Mathematical Society, Providence, RI, 1969.Google Scholar
Helton, J. W. and Howe, R. E., Traces of commutators of integral operators . Acta Math. 135(1975), nos. 3–4, 271–305.CrossRefGoogle Scholar
Koplienko, L. S., Trace formula for perturbations of nonnuclear type . Sib. Mat. Zh. 25(1984), no. 5, 62–71 (Russian). English transl. in Siberian Math. J., 25 (1984), 735–743.Google Scholar
Kreĭn, M. G., On the trace formula in perturbation theory . Mat. Sbornik N.S. 33(1953), no. 75, 597–626. (Russian).Google Scholar
Kreĭn, M. G., On perturbation determinants and a trace formula for unitary and self-adjoint operators. Dokl. Akad. Nauk SSSR 144(1962), 268–271 (Russian).Google Scholar
Langer, H., Eine Erweiterung der Spurformel der Störungstheorie . Math. Nachr. 30(1965) 123–135.CrossRefGoogle Scholar
Lifšic, I. M., On a problem of the theory of perturbations connected with quantum statistics . Uspehi Matem. Nauk (N.S.) 7(1952), no. 1, 171–180 (Russian).Google Scholar
Marcantognini, S. A. M. and Morán, M. D., Koplienko-Neidhardt trace formula for pairs of contractions and pairs of maximal dissipative operators . Math. Nachr. 279(2006), 784–797.CrossRefGoogle Scholar
Malamud, M. M., Neidhardt, H., and Peller, V. V., Absolute continuity of spectral shift . J. Funct. Anal. 276(2019), no. 5, 1575–1621.10.1016/j.jfa.2018.05.011CrossRefGoogle Scholar
Milovanovic, G. V., Mitrinovic, D. S., and Rassias, T. M., Topics in polynomials: Extremal problems inequalities zeros. World Scientific Publishing, River Edge, NJ, 1994.10.1142/1284CrossRefGoogle Scholar
Sz.-Nagy, B., Foias, C., Bercovici, H., and Kérchy, L., Harmonic analysis of operators on Hilbert space, Revised and enlarged edition, Universitext, Springer, New York, NY, 2010.CrossRefGoogle Scholar
Neidhardt, H., Spectral shift function and Hilbert-Schmidt perturbation: extensions of some work of L. S. Koplienko . Math. Nachr. 138(1988), 7–25.CrossRefGoogle Scholar
Neidhardt, H., Scattering matrix and spectral shift of the nuclear dissipative scattering theory. II . J. Oper. Theory 19(1988), no. 1, 43–62.Google Scholar
van Nuland, T. D. H. and Skripka, A., Spectral shift for relative Schatten class perturbations . J. Spectr. Theory 12(2022), no. 4, 1347–1382.CrossRefGoogle Scholar
Peller, V. V., Multiple operator integrals and higher operator derivatives . J. Funct. Anal. 223(2006), 515–544.CrossRefGoogle Scholar
Potapov, D., Skripka, A., and Sukochev, F., Spectral shift function of higher order . Invent. Math. 193(2013), no. 3, 501–538.10.1007/s00222-012-0431-2CrossRefGoogle Scholar
Potapov, D., Skripka, A., and Sukochev, F., Higher-order spectral shift for contractions . Proc. Lond. Math. Soc. (3) 108(2014), no. 2, 327–349.CrossRefGoogle Scholar
Potapov, D., Skripka, A., and Sukochev, F., Trace formulas for resolvent comparable operators . Adv. Math. 272(2015), 630–651.10.1016/j.aim.2014.12.016CrossRefGoogle Scholar
Potapov, D., Skripka, A., and Sukochev, F., Functions of unitary operators: Derivatives and trace formulas. J. Funct. Anal. 270(2016), no. 6, 2048–2072.CrossRefGoogle Scholar
Potapov, D. and Sukochev, F., Koplienko spectral shift function on the unit circle . Commun. Math. Phys. 309(2012), no. 3, 693–702.CrossRefGoogle Scholar
Simon, B., Trace ideals and their applications. 2nd ed., Mathematical Surveys and Monographs, 120, American Mathematical Society, Providence, RI, 2005.Google Scholar
Schäffer, J. J., On unitary dilations of contractions . Proc. Amer. Math. Soc. 6(1955), 322.10.1090/S0002-9939-1955-0068740-7CrossRefGoogle Scholar
Skripka, A., Estimates and trace formulas for unitary and resolvent comparable perturbations . Adv. Math. 311(2017), 481–509.10.1016/j.aim.2017.02.026CrossRefGoogle Scholar
Skripka, A. and Tomskova, A., Multilinear operator integrals. Theory and applications, Lecture Notes in Mathematics, 2250, Springer, Cham, 2019.CrossRefGoogle Scholar
Yosida, K., Functional analysis, Fundamental Principles of Mathematical Sciences, 123, Springer-Verlag, Berlin, 1980.Google Scholar