Koplienko-Neidhardt trace formula for unitaries $-$ A new proof

TL;DR

This paper presents a new proof of the Koplienko-Neidhardt trace formula for unitary operators, simplifying the derivation by reducing it to a finite-dimensional problem, building on previous methods used for Krein's trace formula.

Contribution

It introduces an alternative proof of the Koplienko-Neidhardt trace formula for unitary operators using finite-dimensional reduction techniques.

Findings

New proof simplifies understanding of the trace formula.

Reduces the problem to finite-dimensional case for clarity.

Connects the proof to methods used in Krein's trace formula.

Abstract

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 in 1988, a similar formula was obtained by Neidhardt \cite{NH} in the case of unitary operators. In this article, we give a still another proof of Koplienko-Neidhardt trace formula in the case of unitary operators 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}.