Koplienko spectral shift function on the unit circle
The authors solve in the paper a question raised by \textit{F. Gesztesy}, \textit{A. Pushnitski} and \textit{B. Simon} [``On the Koplienko spectral shift function. I: Basics'', Zh. Mat. Fiz. Anal. Geom. 4, No. 1, 63--107 (2008; Zbl 1190.47016)] regarding a trace formula of Koplienko with summable spectral shift function. It is well known that, for a pair of selfadjoint operators \(H_0\) and \(H_1\) on a Hilbert space with difference \(H_0-H_1\) belonging to the trace class, there exists a uniquely determined function \(\xi_{H_0,H_1}\in L^1\) (the spectral shift function) such that \(\operatorname{Tr}(f(H_1)-f(H_0))=\int_{-\infty}^\infty \xi (\lambda) f'(\lambda)\,d\lambda\) for a certain class of functions \(f\). In the paper under review, a similar formula when \(H_0-H_1\) is Hilbert-Schmidt is considered, and the trace of \(R_2(f, U_0, U_1)= f(U_1)-f(U_0) - \frac{d}{dt} f(U_t)|_{t=0}\), where \(U_t= (1-t)U_0+tU_1\), is analyzed, extending certain results by Koplienko. It is shown that, if \(U_1,U_0\) are contractions such that \(U_1-U_0\) is Hilbert-Schmidt, then there exists \(\eta_{U_0,U_1}\in L^1(\mathbb T)\) such that \(\operatorname{Tr}(R_2(f, U_0, U_1))= \int_{\mathbb T} f''(z)\eta(z)\,dz\) for every polynomial \(f\). Furthermore, for every \(\varepsilon>0\), the function \(\eta\) can be chosen such that \(\|\eta\|_1\leq (1+\varepsilon) \|U_1-U_0\|.\)
scientific article; zbMATH DE number 6006471
scientific article