Related papers: Graph Subspaces and the Spectral Shift Function

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),…

Closed operators in Hilbert space defined by a non-self-adjoint resolution of the identity $\{X(\lambda)\}_{\lambda\in {\mb R}}$, whose adjoints constitute also a resolution of the identity, are studied . In particular, it is shown that a…

We study the spectral theory of operators, generated as direct sums of self-adjoint extensions of quasi-differential minimal operators on a multi-interval set (self-adjoint vector-operators), acting in a Hilbert space. Spectral theorems for…

We give a detailed description of the resolution of the identity of a second order $q$-difference operator considered as an unbounded self-adjoint operator on two different Hilbert spaces. The $q$-difference operator and the two choices of…

In this paper I prove existence of an irreducible pair of operators $H$ and $H+V,$ where $H$ is a self-adjoint operator and $V$ is a self-adjoint trace-class operator, such that the singular spectral shift function of the pair is non-zero…

Given a finite set $X\subseteq\R$ we characterize the diagonals of self-adjoint operators with spectrum $X$. Our result extends the Schur-Horn theorem from a finite dimensional setting to an infinite dimensional Hilbert space analogous to…

We consider non-self-adjoint operators in Hilbert spaces of the form $H=H_0+CWC$, where $H_0$ is self-adjoint, $W$ is bounded and $C$ is a metric operator, $C$ bounded and relatively compact with respect to $H_0$. We suppose that…

In this paper, we report on new results related to the existence of an adjoint for operators on separable Banach spaces and discuss a few interesting applications. (Some results are new even for Hilbert spaces.) Our first two applications…

We provide the mathematical foundation for the $X_m$-Jacobi spectral theory. Namely, we present a self-adjoint operator associated to the differential expression with the exceptional $X_m$-Jacobi orthogonal polynomials as eigenfunctions.…

Given a self-adjoint operator H, a self-adjoint trace class operator V and a fixed Hilbert-Schmidt operator F with trivial kernel and co-kernel, using limiting absorption principle an explicit set of full Lebesgue measure is defined such…

This paper deals with the generalized spectrum of continuously invertible linear operators defined on infinite dimensional Hilbert spaces. More precisely, we consider two bounded, coercive, and self-adjoint operators $\bc{A, B}: V\mapsto…

In this paper spectral theorems for not necessarily continuous normal and self-adjoint random operators on a complex separable Hilbert space are proved.

Assume that $T$ is a self-adjoint operator on a Hilbert space $\mathcal{H}$ and that the spectrum of $T$ is confined in the union $\bigcup_{j\in J}\Delta_j$, $J\subseteq\mathbb{Z}$, of segments $\Delta_j=[\alpha_j,…

We consider operator matrices {\bf H}= (A_0 B_{01} \\ B_{10} A_{1}) with self-adjoint entries A_i, i=0,1, and bounded B_{01}=B_{10}^*, acting in the orthogonal sum {\cal H}={\cal H}_0\oplus{\cal H}_1 of Hilbert spaces {\cal H}_0 and {\cal…

Joint spectra of tuples of operators are subsets in complex projective space. The corresponding tuple of operators can be viewed as an infinite dimensional analog of a determinantal representation of the joint spectrum. We investigate the…

Let $H_0 = -\Delta + V_0(x)$ be a Schroedinger operator on $L_2(\mathbb{R}^\nu),$ $\nu=1,2,$ or 3, where $V_0(x)$ is a bounded measurable real-valued function on $\mathbb{R}^\nu.$ Let $V$ be an operator of multiplication by a bounded…

Let A(x) be a holomorphic family of bounded self-adjoint operators on a separable Hilbert space H and let A(x)_n be the orthogonal compressions of A(x) to the span of first n elements of an orthonormal basis of H. The problem considered…

We consider a J-self-adjoint 2x2 block operator matrix L in the Feshbach spectral case, that is, in the case where the spectrum of one main-diagonal entry is embedded into the absolutely continuous spectrum of the other main-diagonal entry.…

We study finitely cyclic self-adjoint operators in a Hilbert space, i.e. self-adjoint operators that posses such a finite subset in the domain that the orbits of all its elements with respect to the operator are linearly dense in the space.…

In this paper we prove that for an arbitrary pair $\{T_1,T_0\}$ of contractions on Hilbert space with trace class difference, there exists a function $\boldsymbol\xi$ in $L^1({\Bbb T})$ (called a spectral shift function for the pair…