Second‐order trace formulas

Abstract Koplienko Sib. Mat. Zh. 25 (1984), 62–71; English transl. in Siberian Math. J. 25 (1984), 735–743 found a trace formula for perturbations of self‐adjoint operators by operators of Hilbert–Schmidt class . Later, Neidhardt introduced a similar formula in the case of pairs of unitaries via multiplicative path in Math. Nachr. 138 (1988), 7–25. In 2012, Potapov and Sukochev Comm. Math. Phys. 309 (2012), no. 3, 693–702 obtained a trace formula like the Koplienko trace formula for pairs of contractions by answering an open question posed by Gesztesy, Pushnitski, and Simon Zh. Mat. Fiz. Anal. Geom. 4 (2008), no. 1, 63–107, 202; Open Question 11.2. In this paper, we supply a new proof of the Koplienko trace formula in the case of pairs of contractions , where the initial operator 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 Oper. Theory Adv. Appl. 24 (1987) 329–332 and Sinha and Mohapatra Proc. Indian Acad. Sci. Math. Sci. 104 (1994), no. 4, 819–853 and Integral Equations Operator Theory 24 (1996), no. 3, 285–297. 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 pairs of contractions via multiplicative path using the finite‐dimensional approximation method.