The Lifshitz–Krein trace formula and operator Lipschitz functions

Krein’s Trace Formula for Unitary Operators and Operator Lipschitz Functions

The spectral shift function for pairs of selfadjoint operators was introduced in the paper by I.M. Lifshits [17]. In the same paper a trace formula for the difference of functions of the perturbed operator and the unperturbed operator was established. Ideas by Lifshits were developed in the paper by M.G. Krein [14], in which the spectral shift function ξ in L1(R) was defined for arbitrary pairs...

On fully operator Lipschitz functions

Let A(D) be the disc algebra of all continuous complex-valued functions on the unit disc D holomorphic in its interior. Functions from A(D) act on the set of all contraction operators (‖A‖ 1) on Hilbert spaces. It is proved that the following classes of functions from A(D) coincide: (1) the class of operator Lipschitz functions on the unit circle T; (2) the class of operator Lipschitz functions...

Hilbert operator for automorphic L - functions and a local trace formula

Institute for Mathematical Physics Inverse Problem and the Trace Formula for the Hill Operator, Ii Inverse Problem and the Trace Formula for the Hill Operator, Ii

Consider the Hill operator T = ?d 2 =dx 2 + q(x) on L 2 (R); where q 2 L 2 (0; 1) is a 1-periodic real potential and R 1 0 q(x)dx = 0: The spectrum of T is absolutely continuous and consists of intervals separated by gaps n = (a ? n ; a

The Best Constants for Operator Lipschitz Functions on Schatten Classes

Suppose that f is a Lipschitz function on R with ‖f‖Lip ≤ 1. Let A be a bounded self-adjoint operator on a Hilbert space H. Let p ∈ (1,∞) and suppose that x ∈ B(H) is an operator such that the commutator [A, x] is contained in the Schatten class Sp. It is proved by the last two authors, that then also [f(A), x] ∈ Sp and there exists a constant Cp independent of x and f such that ‖[f(A), x]‖p ≤ ...