Koplienko Trace Formula

TL;DR

This paper presents a new proof of Koplienko's trace formula for self-adjoint operators, extending it to unbounded operators by reducing the problem to finite dimensions, building on previous bounded operator results.

Contribution

It introduces a novel proof method for the Koplienko trace formula and extends its applicability to unbounded operators.

Findings

New proof of Koplienko trace formula for bounded operators

Extension of the trace formula to unbounded operators

Reduction to finite-dimensional case for the proof

Abstract

Koplienko gave a trace formula for perturbations of self-adjoint operators by operators of Hilbert-Schmidt class $\mathcal{B}_2(\mathcal{H})$. Recently Gesztesy, Pushnitski and Simon gave an alternative proof of the trace formula when the operators involved are bounded. In this article, we give a still another proof and extend the formula for unbounded case by reducing the problem to a finite dimensional one as in the proof of Krein trace formula by Voiculescu, Sinha and Mohapatra.