Trace formulas for $\mathcal{S}^p$-perturbations and extension of Koplienko-Neidhardt trace formulas
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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866915041783054336 |
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| author | Chattopadhyay, Arup Coine, Clément Giri, Saikat Pradhan, Chandan |
| author_facet | Chattopadhyay, Arup Coine, Clément Giri, Saikat Pradhan, Chandan |
| contents | In this paper, we extend the class of admissible functions for the trace formula of the second order in the self-adjoint, unitary, and contraction cases for a perturbation in the Hilbert-Schmidt class $\mathcal{S}^2(\mathcal{H})$ by assuming a certain factorization of the divided difference $f^{[2]}$. This class is the natural one to ensure that the second order Taylor remainder is a trace class operator. It encompasses all the classes of functions for which the trace formula was previously known. Secondly, for a Schatten $\mathcal{S}^p$-perturbation, $1<p<\infty$, we prove general modified trace formulas for every $n$-times differentiable functions with bounded $n$-th derivative in the self-adjoint and unitary cases and for every $f$ such that $f$ and its derivatives are in the disk algebra $\mathcal{A}(\mathbb{D})$ in the contraction case. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_16426 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Trace formulas for $\mathcal{S}^p$-perturbations and extension of Koplienko-Neidhardt trace formulas Chattopadhyay, Arup Coine, Clément Giri, Saikat Pradhan, Chandan Functional Analysis 47A55, 47A56, 47B10, 47B49 In this paper, we extend the class of admissible functions for the trace formula of the second order in the self-adjoint, unitary, and contraction cases for a perturbation in the Hilbert-Schmidt class $\mathcal{S}^2(\mathcal{H})$ by assuming a certain factorization of the divided difference $f^{[2]}$. This class is the natural one to ensure that the second order Taylor remainder is a trace class operator. It encompasses all the classes of functions for which the trace formula was previously known. Secondly, for a Schatten $\mathcal{S}^p$-perturbation, $1<p<\infty$, we prove general modified trace formulas for every $n$-times differentiable functions with bounded $n$-th derivative in the self-adjoint and unitary cases and for every $f$ such that $f$ and its derivatives are in the disk algebra $\mathcal{A}(\mathbb{D})$ in the contraction case. |
| title | Trace formulas for $\mathcal{S}^p$-perturbations and extension of Koplienko-Neidhardt trace formulas |
| topic | Functional Analysis 47A55, 47A56, 47B10, 47B49 |
| url | https://arxiv.org/abs/2411.16426 |