Second order trace formulae
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2021
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866910396702523392 |
|---|---|
| author | Chattopadhyay, Arup Das, Soma Pradhan, Chandan |
| author_facet | Chattopadhyay, Arup Das, Soma Pradhan, Chandan |
| contents | Koplienko \cite{Ko} found a trace formula for perturbations of self-adjoint operators by operators of Hilbert-Schmidt class $\mathcal{B}_2(\mathcal{H})$. Later, Neidhardt introduced a similar formula in the case of pair of unitaries $(U,U_0)$ via multiplicative path in \cite{NH}. In 2012, Potapov and Sukochev \cite{PoSu} obtained a trace formula like the Koplienko trace formula for pairs of contractions by answering an open question posed by Gesztesy, Pushnitski, and Simon in \cite[Open Question 11.2]{GePu}. In this article, we supply a new proof of the Koplienko trace formula in the case of pair of contractions $(T,T_0)$, where the initial operator $T_0$ is normal, via linear path by reducing the problem to a finite-dimensional one as in the proof of Krein's trace formula by Voiculescu \cite{Voi}, Sinha and Mohapatra \cite{MoSi94,MoSi96}. Consequently, we obtain the Koplienko trace formula for a class of pairs of contractions using the Schäffer matrix unitary dilation. Moreover, we also obtain the Koplienko trace formula for a pair of self-adjoint operators and maximal dissipative operators using the Cayley transform. At the end, we extend the Koplienko-Neidhardt trace formula for a class of pair of contractions $(T,T_0)$ via multiplicative path. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2104_08864 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Second order trace formulae Chattopadhyay, Arup Das, Soma Pradhan, Chandan Functional Analysis 47A55, 47A56, 47A13, 47B10 Koplienko \cite{Ko} found a trace formula for perturbations of self-adjoint operators by operators of Hilbert-Schmidt class $\mathcal{B}_2(\mathcal{H})$. Later, Neidhardt introduced a similar formula in the case of pair of unitaries $(U,U_0)$ via multiplicative path in \cite{NH}. In 2012, Potapov and Sukochev \cite{PoSu} obtained a trace formula like the Koplienko trace formula for pairs of contractions by answering an open question posed by Gesztesy, Pushnitski, and Simon in \cite[Open Question 11.2]{GePu}. In this article, we supply a new proof of the Koplienko trace formula in the case of pair of contractions $(T,T_0)$, where the initial operator $T_0$ is normal, via linear path by reducing the problem to a finite-dimensional one as in the proof of Krein's trace formula by Voiculescu \cite{Voi}, Sinha and Mohapatra \cite{MoSi94,MoSi96}. Consequently, we obtain the Koplienko trace formula for a class of pairs of contractions using the Schäffer matrix unitary dilation. Moreover, we also obtain the Koplienko trace formula for a pair of self-adjoint operators and maximal dissipative operators using the Cayley transform. At the end, we extend the Koplienko-Neidhardt trace formula for a class of pair of contractions $(T,T_0)$ via multiplicative path. |
| title | Second order trace formulae |
| topic | Functional Analysis 47A55, 47A56, 47A13, 47B10 |
| url | https://arxiv.org/abs/2104.08864 |