Higher order spectral shift

Abstract

We construct higher order spectral shift functions, extending the perturbation theory results of M.G. Krein [M.G. Krein, On a trace formula in perturbation theory, Mat. Sb. 33 (1953) 597-626 (in Russian)] and L.S. Koplienko [L.S. Koplienko, Trace formula for perturbations of nonnuclear type, Sibirsk. Mat. Zh. 25 (1984) 62-71 (in Russian); translation in: Trace formula for nontrace-class perturbations, Siberian Math. J. 25 (1984) 735-743] on representations for the remainders of the first and second order Taylor-type approximations of operator functions. The higher order spectral shift functions represent the remainders of higher order Taylor-type approximations; they can be expressed recursively via the lower order (in particular, Krein's and Koplienko's) ones. We also obtain higher order spectral averaging formulas generalizing the Birman-Solomyak spectral averaging formula. The results are obtained in the semi-finite von Neumann algebra setting, with the perturbation taken in the Hilbert-Schmidt class of the algebra.

Taylor Approximations of Operator Functions

“…More detailed discussion of the first order spectral shift function can be found in [12,50,57] and of the second order one in [23]. When a perturbation V is in the Hilbert-Schmidt class S 2 , the higher order spectral shift functions can be expressed via the lower order ones [18,52]. The former are more sensitive to the displacement of the spectrum under perturbation, as demonstrated in [51,53].…”

“…can fail to extend to a measure of finite variation on R n (see [18,Section 4]). This is one of the reasons suggesting that the case n ≥ 3 requires much more delicate (noncommutative) analysis of operator derivatives than the case n < 3.…”

Spectral shift function of higher order

“…Let M ⊆ B(H) be a semifinite von Neumann algebra and τ a normal faithful semifinite trace on M. Let L α denote the noncommutative L α -space with respect to (M, τ) and L α the τ-Schatten-von Neumann ideal L α ∩ M (see, e.g., [2,24] and references cited therein for basic definitions and facts). The existence of η 1 and η 2 satisfying (0.1), where the standard trace Tr is replaced with τ and V is taken from L 1 and L 2 , respectively, is due to [1,8] and [13].…”

Trace formulae for perturbations of class $\boldsymbol{{\boldsymbol S}_m}$

“…In [25] it was shown that for every positive integer m and for every pair .A; K/ of self-adjoint operators with K 2 S m , there exists a unique function Á m in L 1 .R/ such that the following trace formula holds: Note that earlier partial results were obtained in [11], [28], and [29]. In § 6 of this paper we obtain most general trace formulae that include the trace formula for operator Taylor polynomials as a special case.…”