Abstract
We establish estimates and representations for the remainders of Taylor approximations of the spectral action functional \(V\mapsto \tau (f(H_0+V))\) on bounded self-adjoint perturbations, where \(H_0\) is a self-adjoint operator with \(\tau \)-compact resolvent in a semifinite von Neumann algebra and f belongs to a broad set of compactly supported functions including n-times differentiable functions with bounded n-th derivative. Our results significantly extend analogous results in Skripka (J Oper Theory 80(1):113–124, 2018), where f was assumed to be compactly supported and \((n+1)\)-times continuously differentiable. If, in addition, the resolvent of \(H_0\) belongs to the noncommutative \(L^n\)-space, stronger estimates are derived and extended to noncompactly supported functions with suitable decay at infinity.
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Data sharing is not applicable to this article as no data sets were generated or analyzed during the current study.
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Acknowledgements
The research of the first named author is supported by the Mathematical Research Impact Centric Support (MATRICS) grant, File No: MTR/2019/000640, by the Science and Engineering Research Board (SERB), Department of Science & Technology (DST), Government of India. The second named author gratefully acknowledge the support provided by IIT Guwahati, Government of India. The research of the third named author is supported in part by NSF grant DMS-1554456.
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Chattopadhyay, A., Pradhan, C. & Skripka, A. Approximation of the Spectral Action Functional in the Case of \(\tau \)-Compact Resolvents. Integr. Equ. Oper. Theory 95, 20 (2023). https://doi.org/10.1007/s00020-023-02740-9
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DOI: https://doi.org/10.1007/s00020-023-02740-9