Estimates for restrictions of monotone operators on the cone of decreasing functions in Orlicz space

We introduce a number of notions related to the Lyapunov transformation of linear differential operators with unbounded operator coefficients generated by a family of evolution operators. We prove statements about similar operators related to the Lyapunov transformation and describe their spectral properties. One of the main results of the paper is a similarity theorem for a perturbed differential operator with constant operator coefficient, an operator which is the generator of a bounded group of operators. For the perturbation, we consider the operator ofmultiplication by a summable operator function. The almost periodicity (at infinity) of the solutions of the corresponding homogeneous differential equation is established.     

Rank one perturbations of not semibounded operators

Rank one perturbations of selfadjoint operators which are not necessarily semibounded are studied in the present paper. It is proven that such perturbations are uniquely defined, if they are bounded in the sense of forms. We also show that form unbounded rank one perturbations can be uniquely defined if the original operator and the perturbation are homogeneous with respect to a certain one parameter semigroup. The perturbed operator is defined using the extension theory for symmetric operators. The resolvent of the perturbed operator is calculated using Krein's formula. It is proven that every rank one perturbation can be approximated in the operator norm. We prove that some form unbounded perturbations can be approximated in the strong resolvent sense without renormalization of the coupling constant only if the original operator is not semibounded. The present approach is applied to study first derivative and Dirac operators with point interaction, in one dimension.     

On the Similarity of Sturm–Liouville Operators with Non-Hermitian Boundary Conditions to Self-Adjoint and Normal Operators

We consider one-dimensional Schrödinger-type operators in a bounded interval with non-self-adjoint Robin-type boundary conditions. It is well known that such operators are generically conjugate to normal operators via a similarity transformation. Motivated by recent interests in quasi-Hermitian Hamiltonians in quantum mechanics, we study properties of the transformations and similar operators in detail. In the case of parity and time reversal boundary conditions, we establish closed integral-type formulae for the similarity transformations, derive a non-local self-adjoint operator similar to the Schrödinger operator and also find the associated “charge conjugation” operator, which plays the role of fundamental symmetry in a Krein-space reformulation of the problem.     

Error Bounds for Perturbation of the Drazin Inverse of Closed Operators with Equal Spectral Projections

We study perturbations of the Drazin inverse of a closed linear operator A for the case when the perturbed operator has the same spectral projection as A . This theory subsumes results recently obtained by Wei and Wang, Rako ) evi ' and Wei, and Castro and Koliha. We give explicit error estimates for the perturbation of Drazin inverse, and error estimates involving higher powers of the operators.     

Dynamics of hybrid systems induced by operator valued measures

In this paper we consider a class of hybrid systems induced by operator valued measures. This includes semigroups of operators perturbed by bounded as well as unbounded operator valued measures. We construct an evolution operator for the hybrid system and based on its properties we prove existence, uniqueness and regularity properties of solutions. We also consider Semilinear Problems driven by vector measures. Nonstandard problems arising in the study of the classical linear quadratic regulator problem in the present setting are discussed and partial solutions provided.     

Convergence of unilateral problems for monotone operators

We prove some new results on the convergence of variational inequalities for monotone operators, when both the operator and the obstacle are perturbed.     

On the calculation of the spectra of directionally perturbed operators

A class of operators with special spectral properties is defined. An operator in this class is fairly simple, acts in a separable Hilbert space, and can be perturbed so that an a priori given function from its domain is an eigenfunction of the perturbed operator. This fact is shown to be useful for constructing operators in mathematical physics. Specific examples are given.     

Spectral analysis of pseudodifferential operators

The subject of this paper is the spectral analysis of pseudodifferential operators in the framework of perturbation theory. We build up a closed extension (the closure, or the Friedrichs extension) of the perturbed operator. We also prove Weyl-type theorems on the invariance of the essential spectrum of the unperturbed operator. In the case when the perturbed operator is symmetric we obtain a self-adjoint extension. Finally, we consider the case of the relativistic, spin-zero Hamiltonian, with a large class of interactions containing both local potentials, like the Coulomb and Yukawa, and nonlocal ones.     

Perturbation of the Drazin inverse for closed linear operators

We investigate the perturbation of the Drazin inverse of a closed linear operator recently introduced by second author and Tran, and derive explicit bounds for the perturbations under certain restrictions on the perturbing operators. We give applications to the solution of perturbed linear equations, to the asymptotic behaviour ofC ₀-semigroups of linear operators, and to perturbed differential equations. As a special case of our results we recover recent perturbation theorems of Wei and Wang.     

Spectral analysis and solvability of abstract hyperbolic equations with aftereffect

We analyze functional-differential equations with unbounded operator coefficients in a Hilbert space whose leading part is an abstract hyperbolic equation perturbed by terms with a retarded argument and by terms with Volterra integral operators.We consider spectral problems for the operator functions that are the symbols of abovementioned equations in the autonomous case.     

Well-defined solvability and spectral properties of abstract hyperbolic equations with aftereffect

We study functional differential equations with unbounded operator coefficients in Hilbert spaces such that the principal part of the equation is an abstract hyperbolic equation perturbed by terms with delay and terms containing Volterra integral operators. The well-posed solvability of initial boundary-value problems for the specified problems in weighted Sobolev spaces on the positive semi-axis is established.     

Perturbazioni singolari ellittiche

We consider general boundary value problem for partial differential operators with small parameter ε in their coefficients, so-called singular perturbation. Both the perturbed and reduced (with ε=0) problems are supposed to be elliptic and satisfy the Shapiro-Lopatinsky coerciveness condition (see [9], [13]). We point out necessary and sufficient conditions on the operator in the region and the boundary operators for the singulary perturbed boundary value problem to be coercive, i.e. for a characteristic two-sided a priori estimate to hold for its solutions uniformly with respect to ε.     

Singularly perturbed normal operators

We present a generalization of the definition of singularly perturbed operators to the case of normal operators. To do this, we use the idea of normal extensions of a prenormal operator and prove the relation for resolvents of normal extensions similar to the M. Krein relation for resolvents of self-adjoint extensions. In addition, we establish a one-to-one correspondence between the set of singular perturbations of rank one and the set of singularly perturbed (of rank one) operators. Kiev Polytechnic Institute, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 8, pp. 1045–1053, August, 1999.     

A gap in the spectrum of the Neumann–Laplacian on a periodic waveguide

We study a spectral problem related to the Laplace operator in a singularly perturbed periodic waveguide. The waveguide is a quasi-cylinder which contains a periodic arrangement of inclusions. On the boundary of the waveguide, we consider both Neumann and Dirichlet conditions. We prove that provided the diameter of the inclusion is small enough the spectrum of Laplace operator contains band gaps, i.e. there are frequencies that do not propagate through the waveguide. The existence of the band gaps is verified using the asymptotic analysis of elliptic operators.     

On a method of defining the Fourier transform of nonselfadjoint operators

We give a generalization of the study of nonselfadjoint perturbations of selfadjoint operators and the Fourier transforms of such perturbations to the case when the unperturbed operator is nonselfadjoint and has a simple structure. In defining and studying the Fourier transform of the perturbed operator we use smoothness of Kato type of the perturbation with respect to the unperturbed operator.Translated fromMatematicheskie Melody i Fiziko-Mekhanicheskie Polya, Issue 34, 1991, pp. 19–21.     

Covering Results and Perturbed Roper–Suffridge Operators

This work is devoted to the advanced study of Roper–Suffridge type extension operators. For a given non-normalized spirallike function (with respect to an interior or boundary point) on the open unit disk of the complex plane, we construct perturbed extension operators in a certain class of Banach spaces and prove that these operators preserve the spirallikeness property. In addition, we present an extension operator for semigroup generators. We use a new geometric approach based on the connection between spirallike mappings and one-parameter continuous semigroups. It turns out that the new one-dimensional covering results established below are crucial for our investigation.     

Relative oscillation theory for Dirac operators

We develop relative oscillation theory for one-dimensional Dirac operators which, rather than measuring the spectrum of one single operator, measures the difference between the spectra of two different operators. This is done by replacing zeros of solutions of one operator by weighted zeros of Wronskians of solutions of two different operators. In particular, we show that a Sturm-type comparison theorem still holds in this situation and demonstrate how this can be used to investigate the number of eigenvalues in essential spectral gaps. Furthermore, the connection with Krein's spectral shift function is established. As an application we extend a result by K.M. Schmidt on the finiteness/infiniteness of the number of eigenvalues in essential spectral gaps of perturbed periodic Dirac operators.     

New approximation-solvability of general nonlinear operator inclusion couples involving (A, η, m)-resolvent operators and relaxed cocoercive type operators

In this paper, we consider and study a class of general nonlinear operator inclusion couples involving (A, η, m)-resolvent operators and relaxed cocoercive type operators in Hilbert spaces. We also construct a new perturbed iterative algorithm framework with errors and investigate variational graph convergence analysis for this algorithm framework in the context of solving the nonlinear operator inclusion couple along with some results on the resolvent operator corresponding to (A, η, m)-maximal monotonicity. The obtained results improve and generalize some well known results in recent literatures.     

Trace formula for a perturbed operator of Jacobi type

There is a broad class of problems of mathematical physics that lead to the solution of second-order differential equations of some special form. In particular, systems of solutions of such equations are given by classical polynomials (Jacobi, Laguerre, and Hermite polynomials). Such equations are naturally related to second-order differential operators in appropriate Hilbert spaces and the corresponding spectral problems. We consider a Jacobi operator and its perturbation by the operator of multiplication by a function. We derive a trace formula for the perturbed operator and a closed-form expression for the first correction.     

Closed Semistable Operators and Singular Differential Equations

We study a class of closed linear operators on a Banach space whose nonzero spectrum lies in the open left half plane, and for which 0 is at most a simple pole of the operator resolvent. Our spectral theory based methods enable us to give a simple proof of the characterization of C ₀-semigroups of bounded linear operators with asynchronous exponential growth, and recover results of Thieme, Webb and van Neerven. The results are applied to the study of the asymptotic behavior of the solutions to a singularly perturbed differential equation in a Banach space.