Second order trace formulae

Abstract

Koplienko [16] found a trace formula for perturbations of self-adjoint operators by operators of Hilbert-Schmidt class B2(H). In 2012, Potapov and Sukochev [33] obtained a similar formula for pairs of contractions by answering an open question posed by Gesztesy, Pushnitski, and Simon in [13, Open Question 11.2]. 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 [43], Sinha and Mohapatra [23,24]. 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.