Related papers: Real-valued spectral shift functions for contracti…

For the pair $\{-\Delta, -\Delta-\alpha\delta_\mathcal{C}\}$ of self-adjoint Schr\"{o}dinger operators in $L^2(\mathbb{R}^n)$ a spectral shift function is determined in an explicit form with the help of (energy parameter dependent)…

We study the Koplienko Spectral Shift Function (KoSSF), which is distinct from the one of Krein (KrSSF). KoSSF is defined for pairs $A,B$ with $(A-B)\in\calI_2$, the Hilbert-Schmidt operators, while KrSSF is defined for pairs $A,B$ with…

We compute the Fredholm index, ${\rm ind}(D_A)$, of the operator $D_A = (d/dt) + A$ on $L^2(\mathbb{R};\mathcal{H})$ associated with the operator path $\{A(t)\}_{t=-\infty}^{\infty}$, where $(A f)(t) = A(t) f(t)$ for a.e. $t\in\mathbb{R}$,…

We derive explicit Krein resolvent identities for generally singular Sturm-Liouville operators in terms of boundary condition bases and the Lagrange bracket. As an application of the resolvent identities obtained, we compute the trace of…

One of the long standing questions in the theory of Schatten-von Neumann ideals of compact operators is whether their norms have the same differentiability properties as the norms of their commutative counterparts. We answer this question…

We discuss applications of the M. G. Kre\u{\i}n theory of the spectral shift function to the multi-dimensional Schr\"odinger operator as well as specific properties of this function, for example, its high-energy asymptotics. Trace…

We derive two main results: First, assume that $A$, $B$, $A_n$, $B_n$ are self-adjoint operators in the Hilbert space $\mathcal{H}$, and suppose that $A_n$ converges to $A$ and $B_n$ to $B$ in strong resolvent sense as $n \to \infty$. Fix…

We present a new and simple approach to the theory of multiple operator integrals that applies to unbounded operators affiliated with general von Neumann algebras. For semifinite von Neumann algebras we give applications to the Fr\'echet…

Spectral properties of many finite convolution integral operators have been understood by finding differential operators that commute with them. In this paper we compile a complete list of such commuting pairs, extending previous work to…

In this article, we provide the spectral analysis of a Dirac-type operator on $\mathbb{Z}^2$ by describing the behavior of the spectral shift function associated with a sign-definite trace-class perturbation by a multiplication operator. We…

We prove the long standing conjecture in the theory of two-point boundary value problems that completeness and Dunford's spectrality imply Birkhoff regularity. In addition we establish the even order part of S.G.Krein's conjecture that…

The spectral operator was introduced by M. L. Lapidus and M. van Frankenhuijsen [La-vF3] in their reinterpretation of the earlier work of M. L. Lapidus and H. Maier [LaMa2] on inverse spectral problems and the Riemann hypothesis. In…

In (J. Funct. Anal. 257, 1092-1132 (2009)), Dykema and Skripka showed the existence of higher order spectral shift functions when the unperturbed self-adjoint operator is bounded and the perturbations is Hilbert-Schmidt. In this article, we…

We start with the Birman--Solomyak approach to define double operator integrals and consider applications in estimating operator differences $f(A)-f(B)$ for self-adjoint operators $A$ and $B$. We present the Birman--Solomyak approach to the…

Koplienko \cite{Ko} found a trace formula for perturbations of self-adjoint operators by operators of Hilbert-Schmidt class $\mathcal{B}_2(\mathcal{H})$. Later, Neidhardt introduced a similar formula in the case of pair of unitaries…

We explore connections between Krein's spectral shift function $\xi(\lambda,H_0,H)$ associated with the pair of self-adjoint operators $(H_0,H)$, $H=H_0+V$ in a Hilbert space $\calH$ and the recently introduced concept of a spectral shift…

We develop relative oscillation theory for one-dimensional Dirac operators which, rather than measuring the spectrum of one single operator, measures the difference between the spectra of two different operators. This is done by replacing…

We consider contractive operators $T$ that are trace class perturbations of a unitary operator $U$. We prove that the dimension functions of the absolutely continuous spectrum of $T$, $T^*$ and of $U$ coincide. In particular, if $U$ has a…

This is a survey article. We consider different problems in connection with the behavior of functions of operators under perturbations of operators. We deal with three classes of operators: unitary operators, self-adjoint operators, and…

We obtain a representation formula for the derivative of the spectral shift function $\xi(\lambda; B, \epsilon)$ related to the operators $H_0(B,\epsilon) = (D_x - By)^2 + D_y^2 + \epsilon x$ and $H(B, \epsilon) = H_0(B, \epsilon) + V(x,y),…