In the first section we provide a solution to the M. G. Krein problem about an inner description of the space $L_2(Σ,H).$
In the second section we introduce the multiplicity function for an operator measure. Making use of the description of the space $L_2(Σ,H)$ we establish the correctness of the definition and give a criterion for a spectral measure to be a dilation of a given operator measure.
In the third section we prove that the set of principal vectors of an operator measure is an everywhere dense $G_δ.$ This implies, in particular, that there are a lot of principal vectors in any cyclic subspace of a selfadjoint operator.
In the 4th section we introduce Hellinger spectral types for an arbitrary operator measure and prove the existence of subspaces, realizing them.
In the 5-th section we give an answer to an old question of Paulsen and provide an analytic description of completely bounded operator measures. The results of this section belong to the second author.
We establish spectral inclusion and mapping theorems for scalar type spectral operators, generalizing their counterparts for normal operators. Thereby, we extend a precise weak spectral mapping theorem, known to hold for $C_0$-semigroups of normal operators on complex Hilbert spaces, to the more general case of $C_0$-semigroups of scalar type spectral operators on complex Banach spaces. The finer spectrum structure is given itemized consideration.
We derive necessary and sufficient conditions for a Hill operator (i.e., a one-dimensional periodic Schrödinger operator) $H=-d^2/dx^2+V$ to be a spectral operator of scalar type. The conditions show the remarkable fact that the property of a Hill operator being a spectral operator is independent of smoothness (or even analyticity) properties of the potential $V$. In the course of our analysis we also establish a functional model for periodic Schrödinger operators that are spectral operators of scalar type and develop the corresponding eigenfunction expansion. The problem of deciding which Hil…
In this paper we develop a version of spectral theory for bounded linear operators on topological vector spaces. We show that the Gelfand formula for spectral radius and Neumann series can still be naturally interpreted for operators on topological vector spaces. Of course, the resulting theory has many similarities to the conventional spectral theory of bounded operators on Banach spaces, though there are several important differences. The main difference is that an operator on a topological vector space has several spectra and several spectral radii, which fit a well-organized pattern.
In this note the notions of trace compatible operators and infinitesimal spectral flow are introduced. We define the spectral shift function as the integral of infinitesimal spectral flow. It is proved that the spectral shift function thus defined is absolutely continuous and Krein's formula is established. Some examples of trace compatible affine spaces of operators are given.
In the smooth scattering theory framework, we consider a pair of self-adjoint operators $H_0$, $H$ and discuss the spectral projections of these operators corresponding to the interval $(-\infty,λ)$. The purpose of the paper is to study the spectral properties of the difference $D(λ)$ of these spectral projections. We completely describe the absolutely continuous spectrum of the operator $D(λ)$ in terms of the eigenvalues of the scattering matrix $S(λ)$ for the operators $H_{0}$ and $H$. We also prove that the singular continuous spectrum of the operator $D(λ)$ is empty.
We describe a new representation of Hankel operators $H$ as pseudo-differential operators $A$ in the space of functions defined on the whole axis. The amplitudes of such operators $A$ have a very special structure: they are products of functions of a one variable only. This representation has numerous spectral consequences both for compact Hankel operators and for operators with the continuous spectrum.
We consider the direct and inverse spectral problems for Dirac operators on $(0,1)$ with matrix-valued potentials whose entries belong to $L_p(0,1)$, $p\in[1,\infty)$. We give a complete description of the spectral data (eigenvalues and suitably introduced norming matrices) for the operators under consideration and suggest a method for reconstructing the potential from the corresponding spectral data.
Let $u=\int_{-\infty}^{+\infty}λdE_λ$ be a self-adjoint operator in a Hilbert space $H$. Our purpose is to provide a non-standard description of the spectral family $(E_λ)$ and the generalized Gelfand eigenvectors.
In this paper we investigate the spectral expansion for the asymptotically spectral differential operators generated in all real line by ordinary differential expression of arbitrary order with periodic matrix-valued coefficients
First we study the spectral singularity at infinity and investigate the connections of the spectral singularities and the spectrality of the Hill operator. Then we consider the spectral expansion when there is not the spectral singularity at infinity.
In this paper we investigate the spectral expansion for the one-dimensional Schrodinger operator with a periodic complex-valued potential. For this we consider in detail the spectral singularities and introduce new concepts as essential spectral singularities and singular quasimomenta.
We study spectral properties of a class of global infinite order pseudo-differential operators and obtain the asymptotic behaviour of the spectral counting functions of such operators. Unlike their finite order counterparts, their spectral asymptotics are not of power-log-type but of log-type. The ultradistributional setting of such operators of infinite order makes the theory more complex so that the standard finite order global Weyl calculus cannot be used in this context.
Using the method of similar operators we study an even order differential operator with periodic, semiperiodic, and Dirichlet boundary conditions. We obtain asymptotic formulas for eigenvalues of this operator and estimates for its spectral decompositions and spectral projections. We also establish the asymptotic behavior of the corresponding analytic semigroup of operators.
We introduce the concept of a spectral shift operator and use it to derive Krein's spectral shift function for pairs of self-adjoint operators. Our principal tools are operator-valued Herglotz functions and their logarithms. Applications to Krein's trace formula and to the Birman-Solomyak spectral averaging formula are discussed.
The paper concerns the spectral theory for a class of non-self-adjoint block convolution operators. We mainly discuss the spectral representations of such operators. It is considered the general case of operators defined on Banach spaces. The main results are applied to periodic Jacobi matrices.
We study spectral properties of nonselfadjoint rank one perturbations of compact selfadjoint operators. The problems under consideration include completeness of eigenvectors, relations between completeness of the perturbed operator and its adjoint, and the spectral synthesis problem. We obtain new criteria for completeness and spectral synthesis in this class as well as a series of counterexamples which show that the spectral structure of rank one perturbations is, in general, unexpectedly rich and complicated. A parallel spectral theory is developed for one-dimensional singular perturbations …
For the relations generated by pair of differential operator expressions one of which depends on the spectral parameter in the Nevanlinna manner we construct analogs of the generalized resolvents which are integro-differential operators. The expansions in eigenfunctions of these relations are obtained.
In this paper we discuss some spectral invariance results for non-smooth pseudodifferential operators with coefficients in Hölder spaces. In analogy to the proof in the smooth case of Beals and Ueberberg, we use the characterization of non-smooth pseudodifferential operators to get such a result. The main new difficulties are the limited mapping properties of pseudodifferential operators with non-smooth symbols and the fact, that in general the composition of two non-smooth pseudodifferential operators is not a pseudodifferential operator. In order to improve these spectral invariance results …
Important spectral features, such as the emptiness of the residual spectrum, countability of the point spectrum, provided the space is separable, and a characterization of spectral gap at $0$, known to hold for bounded scalar type spectral operators, are shown to naturally transfer to the unbounded case.
In this paper we investigate the non-self-adjoint operator H generated in all real line by the Mathieu-Hill equation with a complex-valued potential. We find a necessary and sufficient conditions on the potential for which H has no spectral singularity at infinity and it is an asymptotically spectral operator. Moreover, we give a detailed classification, stated in term of the potential, for the form of the spectral decomposition of the operator H by investigating the essential spectral singularities.
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 such operators are discussed, the structure of the ordered spectral representation is investigated for the case of differential coordinate operators. One of the main results is the construction of spectral resolutions. Finally, we study the matters connected with analytical decompositions of generalized eigenfunctions of such vector-operators and build a ma…
First order integro-differential operators on a finite interval are studied. Properties of spectral characteristic are established, and the uniqueness theorem is proved for the inverse problem of recovering operators from their spectral data.
The recently introduced concept of a spectral shift operator is applied in several instances. Explicit applications include Krein's trace formula for pairs of self-adjoint operators, the Birman-Solomyak spectral averaging formula and its operator-valued extension, and an abstract approach to trace formulas based on perturbation theory and the theory of self-adjoint extensions of symmetric operators.
In this paper spectral theorems for not necessarily continuous normal and self-adjoint random operators on a complex separable Hilbert space are proved.
Inverse spectral problems are studied for the second order integro-differential operators on a finite interval. Properties of spectral characteristic are established, and the uniqueness theorem is proved for this class of inverse problems.
In the paper, Sturm--Liouville differential operators on time scales consisting of a finite number of isolated points and segments are considered. Such operators unify differential and difference operators. We obtain properties of their spectral characteristics including asymptotic formulae for eigenvalues and weight numbers. Uniqueness theorem is proved for recovering the operators from the spectral characteristics.
In this paper we investigate the spectrum and spectrality of the one-dimensional Schrodinger operator with a periodic PT-symmetric complex-valued potential.
In this paper we prove the spectral theorem for quaternionic unbounded normal operators using the notion of $S$-spectrum. The proof technique consists of first establishing a spectral theorem for quaternionic bounded normal operators and then using a transformation which maps a quaternionic unbounded normal operator to a quaternionic bounded normal operator. With this paper we complete the foundation of spectral analysis of quaternionic operators. The $S$-spectrum has been introduced to define the quaternionic functional calculus but it turns out to be the correct object also for the spectral …
We give a spectral description of the semi-classical Schrodinger operator with a piecewise linear, complex valued potential. Moreover, using these results, we show how an arbitrarily small bounded perturbation of a non-self-adjoint operator can completely change the spectrum of the operator.
We study fractal dimension properties of singular Jacobi operators. We prove quantitative lower spectral/quantum dynamical bounds for general operators with strong repetition properties and controlled singularities. For analytic quasiperiodic Jacobi operators in the positive Lyapunov exponent regime, we obtain a sharp arithmetic criterion of full spectral dimensionality. The applications include the extended Harper's model where we obtain arithmetic results on spectral dimensions and quantum dynamical exponents.
In this paper we provide new asymptotic estimates of various spectral quantities of Zakharov-Shabat operators on the circle. These estimates are uniform on bounded subsets of potentials in Sobolev spaces.
Necessary and sufficient conditions are presented for a positive measure to be the spectral measure of a half-line Schrodinger operator with square integrable potential.
Exact Hausdorff dimensions are computed for singular continuous components of the spectral measures of a class of Schrödinger operators in bounded intervals.
We study Sturm-Liouville operators on closed sets of a special structure, which are sometimes referred as time scales and often appear in modelling various real processes. Depending on the set structure, such operators unify both differential and difference operators. We obtain properties of their spectral characteristics and study inverse problems with respect to them. We prove that the spectral characteristics uniquely determine the operator.
We solve the inverse spectral problem of recovering the singular potentials $q\in W^{-1}_{2}(0,1)$ of Sturm-Liouville operators by two spectra. The reconstruction algorithm is presented and necessary and sufficient conditions on two sequences to be spectral data for Sturm-Liouville operators under consideration are given.
We consider Schrödinger operators on periodic discrete graphs. It is known that the spectrum of these operators has band structure. We obtain a localization of spectral bands in terms of eigenvalues of Dirichlet and Neumann operators on a finite graph, which is constructed from the fundamental cell of the periodic graph. The proof is based on the Floquet decomposition of Schrödinger operators and the minimax principle.
We consider the self-adjoint fourth-order operator with real $1$-periodic coefficients on the unit interval. The spectrum of this operator is discrete. We determine the high energy asymptotics for its eigenvalues.
Given two possibly unbounded selfadjoint operators A and G such that the resolvent sets of AG and GA are non-empty, it is shown that the operator AG has a spectral function on IR with singularities if there exists a non-zero polynomial p such that the symmetric operator Gp(AG) is non-negative. This result generalizes a well-known theorem for definitizable operators in Krein spaces.
In this paper we construct the spectral expansion for the non-self-adjoint differential operators generated in the space of vektor functions by the ordinary differential expression of arbitrary order with the periodic matrix coefficients by using the essential spectral singularities, singular quasimomenta and the series with parenthesis.
It is shown that transfer functions, which play a crucial role in M.G. Krein's study of inverse spectral problems, are a proper tool to formulate local spectral uniqueness conditions.
We prove new spectral enclosures for the non-real spectrum of a class of $2\times2$ block operator matrices with self-adjoint operators $A$ and $D$ on the diagonal and operators $B$ and $-B^*$ as off-diagonal entries. One of our main results resembles Gershgorin's circle theorem. The enclosures are applied to $J$-frame operators.
The matrix Sturm-Liouville operator on a finite interval with singular potential of class $W_2^{-1}$ and the general self-adjoint boundary conditions is studied. This operator generalizes the Sturm-Liouville operators on geometrical graphs. We investigate the inverse problem that consists in recovering the considered operator from the spectral data (eigenvalues and weight matrices). The inverse problem is reduced to a linear equation in a suitable Banach space, and a constructive algorithm for the inverse problem solution is developed. Moreover, we obtain the spectral data characterization for…
We characterize all bounded Hankel operators $Γ$ such that $Γ^*Γ$ has finite spectrum. We identify spectral data corresponding to such operators and construct inverse spectral theory including the characterization of these spectral data.
We introduce a notion of $β$-almost periodicity and prove quantitative lower spectral/quantum dynamical bounds for general bounded $β$-almost periodic potentials. Applications include a sharp arithmetic criterion of full spectral dimensionality for analytic quasiperiodic Schrödinger operators in the positive Lyapunov exponent regime and arithmetic criteria for families with zero Lyapunov exponents, with applications to Sturmian potentials and the critical almost Mathieu operator.
The self-adjoint matrix Sturm-Liouville operator on a finite interval with a boundary condition in the general form is studied. We obtain asymptotic formulas for the eigenvalues and the weight matrices of the considered operator. These spectral characteristics play an important role in the inverse spectral theory. Our technique is based on analysis of analytic functions and on the contour integration in the complex plane of the spectral parameter. In addition, we adapt the obtained asymptotic formulas to the Sturm-Liouville operators on a star-shaped graph with two different types of matching …
We prove that the inverse spectral mapping reconstructing the square integrable potentials on [0,1] of Dirac operators in the AKNS form from their spectral data (two spectra or one spectrum and the corresponding norming constants) is analytic and uniformly stable in a certain sense.
The inverse spectral problems are studied for the Sturm-Liouville operator on the star-shaped graph and for the matrix Sturm-Liouville operator with the boundary condition in the general self-adjoint form. We obtain necessary and sufficient conditions of solvability for these two inverse problems, and also prove their local solvability and stability.
In this paper we construct the spectral expansion for the differential operator generated in all real line by ordinary differential expression of arbitrary order with periodic complex-valued coefficients by introducing new concepts as essential spectral singularities and singular quasimomenta and using the series with parenthesis. Moreover, we find a criteria for which the spectral expansion coincides with the Gelfand expansion for the self-adjoint case.
Sign type spectra are an important tool in the investigation of spectral properties of selfadjoint operators in Krein spaces. It is our aim to show that also sign type spectra for normal operators in Krein spaces provide insight in the spectral nature of the operator: If the real part and the imaginary part of a normal operator in a Krein space have real spectra only and if the growth of the resolvent of the imaginary part (close to the real axis) is of finite order, then the normal operator possesses a local spectral function defined for Borel subsets of the spectrum which belong to positive …
