Eigenvalues and Entropy Moduli of Operators in Interpolation Spaces

The paper approaches in an abstract way the spectral theory of operators in abstract interpolation spaces. We introduce entropy numbers and spectral moduli of operators, and prove a relationship between them and eigenvalues of operators. We also investigate interpolation variants of the moduli, and offer a contribution to the theory of eigenvalues of operators. Specifically, we prove an interpolation version of the celebrated Carl–Triebel eigenvalue inequality. Based on these results we are able to prove interpolation estimates for single eigenvalues as well as for geometric means of absolute values of the first n eigenvalues of operators. In particular, some of these estimates may be regarded as generalizations of the classical spectral radius formula. We give applications of our results to the study of interpolation estimates of entropy numbers as well as of the essential spectral radius of operators in interpolation spaces.     

On a class of completely continuous operators in Hilbert spaces

We introduce a class G of completely continuous operators and prove theorems on the spectral structure of these operators. In particular, operators of this class are similar to model operators in de Branges spaces.     

Asymptotic and Spectral Analysis of the Spatially Nonhomogeneous Timoshenko Beam Model

We develop spectral and asymptotic analysis for a class of nonselfadjoint operators which are the dynamics generators for the systems governed by the equations of the spatially nonhomogeneous Timoshenko beam model with a 2–parameter family of dissipative boundary conditions. Our results split into two groups. We prove asymptotic formulas for the spectra of the aforementioned operators (the spectrum of each operator consists of two branches of discrete complex eigenvalues and each branch has only two points of accumulation: +∞ and —∞), and for their generalized eigenvectors. Our second main result is the fact that these operators are Riesz spectral. To obtain this result, we prove that the systems of generalized eigenvectors form Riesz bases in the corresponding energy spaces. We also obtain the asymptotics of the spectra and the eigenfunctions for the nonselfadjoint polynomial operator pencils associated with these operators. The pencil asymptotics are essential for the proofs of the spectral results for the aforementioned dynamics generators.     

Reconstruction of Sturm-Liouville differential operators on A-graphs

We study the inverse problem of spectral analysis for Sturm-Liouville operators on A-graphs. We obtain a constructive procedure for solving the inverse problem of reconstruction of coefficients of differential operators from spectra and prove the uniqueness of the solution.     

Sparse potentials with fractional Hausdorff dimension

We construct non-random bounded discrete half-line Schrödinger operators which have purely singular continuous spectral measures with fractional Hausdorff dimension (in some interval of energies). To do this we use suitable sparse potentials. Our results also apply to whole line operators, as well as to certain random operators. In the latter case we prove and compute an exact dimension of the spectral measures.     

Spectral radius inequalities for Hilbert space operators

We prove several spectral radius inequalities for sums, products, and commutators of Hilbert space operators. Pinching inequalities for the spectral radius are also obtained.

     

Commuting Differential Operators of Rank 2 with Rational Coefficients

In this paper we find new pairs of self-adjoint commuting differential operators of rank 2 with rational coefficients and prove that any curve of genus 2 written as a hyperelliptic curve is the spectral curve of a pair of commuting differential operators with rational coefficients. We also study the case where curves of genus 3 are the spectral curves of pairs of commuting differential operators of rank 2 with rational coefficients.     

Spectral analysis of singular ordinary differential operators with indefinite weights

In this paper we develop a perturbation approach to investigate spectral problems for singular ordinary differential operators with indefinite weight functions. We prove a general perturbation result on the local spectral properties of selfadjoint operators in Krein spaces which differ only by finitely many dimensions from the orthogonal sum of a fundamentally reducible operator and an operator with finitely many negative squares. This result is applied to singular indefinite Sturm-Liouville operators and higher order singular ordinary differential operators with indefinite weight functions.     

Perturbation of spectra and spectral subspaces

We consider the problem of variation of spectral subspaces for linear self-adjoint operators with emphasis on the case of off-diagonal perturbations. We prove a number of new optimal results on the shift of the spectrum and obtain (sharp) estimates on the norm of the difference of two spectral projections associated with isolated parts of the spectrum of the perturbed and unperturbed operators, respectively.

     

Self-Adjoint Operators Generated by Problems of Transmission with Inhomogeneous Conjugation Conditions

For problems of transmission for higher-order equations with inhomogeneous conjugation conditions, we construct and investigate the corresponding operators in Hilbert spaces. We prove the self-adjointness of these operators for a spectral problem with a parameter in conjugation conditions.     

Weighted restriction type estimates for Grushin operators and application to spectral multipliers and Bochner–Riesz summability

We prove weighted restriction type estimates for Grushin operators. These estimates are then used to prove sharp spectral multiplier theorems as well as Bochner–Riesz summability results with sharp exponent.     

Sturm-Liouville operators in the half axis with local perturbations

We give conditions which imply equivalence of the Lebesgue measure with respect to a measure μ generated as an average of spectral measures corresponding to Sturm-Liouville operators in the half axis. We apply this to prove that some spectral properties of these operators hold for large sets of boundary conditions if and only if they hold for large sets of positive local perturbations.     

Absolute continuity of spectral shift

In this paper we develop the method of double operator integrals to prove trace formulae for functions of contractions, dissipative operators, unitary operators and self-adjoint operators. To establish the absolute continuity of spectral shift, we use the Sz.-Nagy theorem on the absolute continuity of the spectral measure of the minimal unitary dilation of a completely nonunitary contraction. We also give a construction of an intermediate contraction for a pair of contractions with trace class difference.     

Semi-classical analysis and pseudo-spectra

We prove an approximate spectral theorem for non-self-adjoint operators and investigate its applications to second-order differential operators in the semi-classical limit. This leads to the construction of a twisted FBI transform. We also investigate the connections between pseudo-spectra and boundary conditions in the semi-classical limit.     

The normed finiteness property of compact contraction operators

We study the joint spectral radius given by a finite set of compact operators on a Hilbert space. It is shown that the normed finiteness property holds in this case, that is, if all the compact operators are contractions and the joint spectral radius is equal to 1 then there exists a finite product that has a spectral radius equal to 1. We prove an additional statement in that the requirement that the joint spectral radius be equal to 1 can be relaxed to the asking that the maximum norm of finite products of a length norm is equal to 1. The length of this product is related to the dimension of the subspace on which the set of operators is norm preserving.     

On some properties of the multidimensional Bochner-Phillips functional calculus

We develop the multidimensional functional calculus of semigroup generators which is based on the class of Bernstein functions in several variables. We establish spectral mapping theorems, give a holomorphy condition for the semigroups generated by the operators arising in this calculus, as well as prove the moment inequality for these operators.     

Spectral analysis of differential operators with unbounded operator coefficients in weighted spaces of functions

The spectral properties of operators constructed from a family of evolution operators are studied in Banach spaces of vector functions defined by a weight function of pre-exponential growth. Necessary and sufficient conditions for the invertibility of such operators are obtained in terms of exponential dichotomy. We prove a spectrum mapping theorem for semigroups of Howland difference operators generated by the operator under study.     

On a class of quasi-Fredholm operators

We study a class of bounded linear operators acting on a Banach spaceX called B-Fredholm operators. Among other things we characterize a B-Fredholm operator as the direct sum of a nilpotent operator and a Fredholm operator and we prove a spectral mapping theorem for B-Fredholm operators.IMemory of my father, Sidi-Bouhouria 1914-0991.     

Log-dimensional spectral properties of one-dimensional quasicrystals

We consider discrete one-dimensional Schrödinger operators on the whole line and establish a criterion for continuity of spectral measures with respect to log-Hausdorff measures. We apply this result to operators with Sturmian potentials and thereby prove logarithmic quantum dynamical lower bounds for all coupling constants and almost all rotation numbers, uniformly in the phase.

     

Averaged wave operators on singular spectrum

We prove the existence of pairs of unitary (or self-adjoint) operators with singular spectral measure whose difference is a rank-two operator for which the Abel wave operators fail to exist. Also, we discuss the closely related problem of constructing the Hilbert transform with respect to a singular measure on the unit circle.