Koplienko trace formula for unitaries via linear path

Arup Chattopadhyay, Soma Das, Chandan Pradhan

Published 2021-04-18Version 1

Koplienko \cite{Ko} found a trace formula for perturbations of self-adjoint operators by operators of Hilbert-Schmidt class $\mathcal{B}_2(\mathcal{H})$. In 2012, Potapov and Sukochev \cite{PoSu} obtained a similar 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 give a still another proof of the Koplienko trace formula in the case of unitary operators 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 Sch$\ddot{a}$ffer 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.