We present a scenario concerning the existence of a large class of reflectionless selfadjoint analytic difference operators. In order to exemplify this scenario, we summarize our results on reflectionless self-adjoint difference operators of relativistic CalogeroMoser type.

Let P be a selfadjoint elliptic operator of order m > 0 acting on the sections of a Hermitian vector bundle over a compact Riemannian manifold of dimension n. General arguments show that its zeta and eta functions may have poles only at points of the form s = k m , where k ranges over all non-zero integers ≤ n. In this paper, we construct elementary and explicit examples of perturbations of P w...

Some vector and operator generalized trapezoidal inequalities for continuous functions of selfadjoint operators in Hilbert spaces are given. Applications for power and logarithmic functions of operators are provided as well.

Some inequalities for convex functions of selfadjoint operators in Hilbert spaces under suitable assumptions for the involved operators are given. Applications for particular cases of interest are also provided.

We study the structure of the Fuč́ık spectra for the linear multipoint differential operators. We introduce a variational approach in order to obtain a robust and global algorithm which is suitable for the exploration of unknown Fuč́ık spectrum structure. We apply our approach in the case of the four-point selfadjoint differential operator of the fourth order which is closely connected to the non...

Spectral points of positive and negative type, and type π+ and type π− for closed linear operators and relations in Krein spaces are introduced with the help of approximative eigensequences. The main objective of the paper is to study these sign type properties in the non-selfadjoint case under various kinds of perturbations, e.g. compact perturbations and perturbations small in the gap metric....

The paper continues the study of differential Banach *algebras AS and FS of operators associated with symmetric operators S on Hilbert spaces H. The algebra AS is the domain of the largest *-derivation δS of B(H) implemented by S and the algebra FS is the closure of the set of all finite rank operators in AS with respect to the norm ‖A‖ = ‖A‖+‖δS(A)‖. When S is selfadjoint, FS is the domain of ...

The multivariate orthogonal polynomials are related to a family of operators whose matrix representations are block Jacobi matrices. A sufficient condition is given so that these operators, in general unbounded, are commuting and selfadjoint. The spectral theorem for these operators is used to establish the existence of the measure of orthogonality in Favard's theorem.

We prove a general operator theoretic result that asserts that many multiplicity two selfadjoint operators have simple singular spectrum. © 2005 Elsevier Inc. All rights reserved.