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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 https://ror.org/0022nd079 , Guwahati 781039, India e-mail: arupchatt@iitg.ac.in, 2003arupchattopadhyay@gmail.com
Chandan Pradhan
Affiliation:
Department of Mathematics, Indian Institute of Science https://ror.org/04dese585 , 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 https://ror.org/05fs6jp91 , Albuquerque, NM 87106, United States
*
e-mail: skripka@math.unm.edu
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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
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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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).

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