Related papers: Koplienko-Neidhardt trace formula for unitaries $-…

We establish higher order trace formulas for pairs of contractions along a multiplicative path generated by a self-adjoint operator in a Schatten-von Neumann ideal, removing earlier stringent restrictions on the kernel and defect operator…

In this paper, we consider an unbounded selfadjoint operator $A$ and its selfadjoint perturbations in the same Hilbert space $\mathcal{H}$. As S.Albeverio and P. Kurosov (2000), we call a selfadjoint operator $A_{1}$ the singular…

A first order trace formula is obtained for a higher-order differential operator on a segment in the case where the perturbation is an operator of multiplication by a finite complex-valued measure. For the operators of even order $n\ge4$ a…

The main result of the paper is that the Lifshits--Krein trace formula cannot be generalized to the case of functions of noncommuting self-adjoint operators. To prove this, we show that for pairs $(A_1,B_1)$ and $(A_2,B_2)$ of bounded…

The aim of this paper is twofold: On one hand we discuss an abstract approach to symmetrized Fredholm perturbation determinants and an associated trace formula for a pair of operators of positive-type, extending a classical trace formula.…

We introduce an appropriate notion of trace in the setting of quaternionic linear operators, arising from the well-known companion matrices. We then use this notion to define the quaternionic Fredholm determinant of trace-class operators in…

We obtain a simple formula for the first-order trace of a regular differential operator on a segment perturbated by a multiplication operator. The main analytic ingredient of the proof is an improvement of the Tamarkin equiconvergence…

An operator $H=H_{0}+V$ where $H_{0}=i^{-N} \partial^{N}$ ($N$ is arbitrary) and $V$ is a differential operator of order $N-1$ with coefficients decaying sufficiently rapidly at infinity is considered in the space $L^2(\Bbb R)$. The goal of…

We study the behaviour of functions of self-adjoint operators under relatively bounded and relatively trace class perturbation We introduce and study the class of relatively operator Lipschitz functions. An essential role is played by…

In this note, we provide an elementary proof for the expression of $f(U)-f(V)$ in the form of a double operator integral for every Lipschitz function $f$ on the unit circle $\cir$ and for a pair of unitary operators $(U,V)$ with…

We investigate trace formulas for Jacobi operators which are trace class perturbations of quasi-periodic finite-gap operators using Krein's spectral shift theory. In particular we establish the conserved quantities for the solutions of the…

For a purely imaginary sign-definite perturbation of a self-adjoint operator, we obtain exponential representations for the perturbation determinant in both upper and lower half-planes and derive respective trace formulas.

The recently introduced concept of a spectral shift operator is applied in several instances. Explicit applications include Krein's trace formula for pairs of self-adjoint operators, the Birman-Solomyak spectral averaging formula and its…

We compute the Selberg trace formula for Hecke operators (also called the trace formula for modular correspondences) in the context of cocompact Kleinian groups with finite-dimentional unitary representations. We give some applications to…

We study the trace class perturbations of the whole-line, discrete Laplacian and obtain a new bound for the perturbation determinant of the corresponding non-self-adjoint Jacobi operator. Based on this bound, we refine the Lieb--Thirring…

Using simple commutator relations, we obtain several trace identities involving eigenvalues and eigenfunctions of an abstract self-adjoint operator acting in a Hilbert space. Applications involve abstract universal estimates for the…

Our goal is to extend the theory of the spectral shift function to the case where only the difference of some powers of the resolvents of self-adjoint operators belongs to the trace class. As an example, we consider a couple of Dirac…

In this note we study the problem of evaluating the trace of $f(T)-f(R)$, where $T$ and $R$ are contractions on Hilbert space with trace class difference, i.e., $T-R\in\boldsymbol{S}_1$ and $f$ is a function analytic in the unit disk ${\Bbb…

We obtain a trace formula for algebraic differential operators which the corresponding analytic results have been proved by M. Engeli and G. Felder

We use the boundary triplet approach to extend the classical concept of perturbation determinants to a more general setup. In particular, we examine the concept of perturbation determinants to pairs of proper extensions of closed symmetric…