Related papers: On Krein's Example

We present a model for spectral theory of families of selfadjoint operators, and their corresponding unitary one-parameter groups (acting in Hilbert space.) The models allow for a scale of complexity, indexed by the natural numbers…

Kadison characterized the diagonals of projections and observed the presence of an integer. Arveson later recognized this integer as a Fredholm index obstruction applicable to any normal operator with finite spectrum coincident with its…

Koplienko gave a trace formula for perturbations of self-adjoint operators by operators of Hilbert-Schmidt class $\mathcal{B}_2(\mathcal{H})$. Recently Gesztesy, Pushnitski and Simon gave an alternative proof of the trace formula when the…

We prove that one-dimensional reflectionless Schr\"odinger operators with spectrum a homogeneous set in the sense of Carleson, belonging to the class introduced by Sodin and Yuditskii, have purely absolutely continuous spectra. This class…

We consider the problem of finding the spectrum of an operator taking the form of a low-rank (rank one or two) non-normal perturbation of a well-understood operator, motivated by a number of problems of applied interest which take this…

We study the spectral convergence of compact, self-adjoint operators on a separable Hilbert space under operator norm perturbations, and derive asymptotic expansions for their eigenvalues and eigenprojections. Our analysis focuses on…

The problem of variation of spectral subspaces for linear self-adjoint operators under an additive bounded perturbation is considered. The aim is to find the best possible upper bound on the norm of the difference of two spectral…

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…

Let $A$ and $A_{1}$ are unbounded selfadjoint operators in a Hilbert space $\mathcal{H}$. Following \cite{AK} we call $A_{1}$ a \textit{singular} perturbation of $A$ if $A$ and $A_{1}$ have different domains…

This paper studies trace class perturbation of closed linear relations in Hilbert spaces. The concept of trace class perturbation of closed relations is introduced by orthogonal projections. Equivalent characterizations of compact and trace…

One of the most important contributions of Heinz Langer in the area of operator theory in Krein spaces is the introduction of the notion of definitizable operators and the construction of the corresponding spectral function. In this note we…

We consider one-dimensional chains and multi-dimensional networks of harmonic oscillators coupled to two Langevin heat reservoirs at different temperatures. Each particle interacts with its nearest neighbors by harmonic potentials and all…

In this paper, we extend Fredholm theory in von Neumann algebras established by Breuer in [5] and [6] to spectral Fredholm theory. We consider 2 by 2 upper triangular operator matrices with coefficients in a von Neumann algebra and give the…

This work presents a contemporary treatment of Krein's entire operators with deficiency indices $(1,1)$ and de Branges' Hilbert spaces of entire functions. Each of these theories played a central role in the research of both renown…

We build a combinatorial invariant, called the spectral monodromy from the spectrum of a non-selfadjoint h -pseudodifferential operator with two degrees of freedom in the semi-classical limit. We treat small non-selfadjoint perturbation of…

We give the first example of a connected 4-regular graph whose Laplace operator's spectrum is a Cantor set, as well as several other computations of spectra following a common ``finite approximation'' method. These spectra are simple…

We study half-line discrete Schr\"odinger operators and their rank-one perturbations. We establish certain continuity and stability properties of the Fourier transform of the associated spectral measures. Using these results, we construct a…

An analogue of the Davis-Kahan sin2\Theta\ theorem from [SIAM J. Numer. Anal. 7 (1970), 1--46] is proved under a general spectral separation condition. This extends the generic sin2\theta\ estimates recently shown by Albeverio and Motovilov…

In an earlier work we had considered a Gaussian ensemble of random matrices in the presence of a given external matrix source. The measure is no longer unitary invariant and the usual techniques based on orthogonal polynomials, or on the…

In this paper we extend the results in [Ra] on the representation of the Hecke algebra, determined by the matrix coefficients of a projective, unitary representation, in the discrete series of representations of the ambient group, to a more…