A posteriori estimates of inverse operators for initial value problems in linear ordinary differential equations

We present constructive a posteriori estimates of inverse operators for initial value problems in linear ordinary differential equations (ODEs) on a bounded interval. Here, “constructive” indicates that we can obtain bounds of the operator norm in which all constants are explicitly given or are represented in a numerically computable form. In general, it is difficult to estimate these inverse operators a priori. We, therefore, propose a technique for obtaining a posteriori estimates by using Galerkin approximation of inverse operators. This type of estimation will play an important role in the numerical verification of solutions for initial value problems in nonlinear ODEs as well as for parabolic initial boundary value problems.     

A posteriori estimates of inverse operators for initial value problems in linear ordinary differential equations

We present constructive a posteriori estimates of inverse operators for initial value problems in linear ordinary differential equations (ODEs) on a bounded interval. Here, “constructive” indicates that we can obtain bounds of the operator norm in which all constants are explicitly given or are represented in a numerically computable form. In general, it is difficult to estimate these inverse operators a priori. We, therefore, propose a technique for obtaining a posteriori estimates by using Galerkin approximation of inverse operators. This type of estimation will play an important role in the numerical verification of solutions for initial value problems in nonlinear ODEs as well as for parabolic initial boundary value problems.     

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.     

Estimates for Elements of Inverse Matrices for a Class of Operators with Matrices of Special Structure

In this paper, we consider questions related to the structure of inverse matrices of linear bounded operators acting in infinite-dimensional complex Banach spaces. We obtain specific estimates of elements of inverse matrices for bounded operators whose matrices have a special structure. Matrices are introduced as special operator-valued functions on an index set. The matrix structure is described by the behavior of the given function on elements of a special partition of the index set. The method used for deriving the estimates is based on an analysis of Fourier series of strongly continuous periodic functions.     

Composition and nuclearity of kernel operators

Let T and S be kernel operators of finite double norm, respectively finite inverse double norm. We prove that the compositions TS and ST are nuclear operators, provided one of the function norms is order continuous. Conversely we prove theorems about factorizations of nuclear kernel operators on Banach function spaces. These results are then used to derive optimal eigenvalue estimates for nuclear operators on Banach lattices by relating Pisier's results with the results of König and Weis.     

Einschließungsaussagen bei Differentialoperatoren zweiter Ordnung durch punktweise Ungleichungen

Summary We consider non-inverse positive differential operators of the second order and show that error estimates can be obtained through methods of inverse positivity alone. This means a significant simplification if the operator has only one negative eigenvalue. In addition we show how to obtain Range-Domain implications for operators with mixed boundary conditions.

The author would like to thank the European Research Office of the United States Army for their kind interest     

On a Result of R. Turner

We study the similarity of perturbed compact operators to operators of block-diagonal structure with respect to some family of orthogonal projection operators, which allows us to refine and essentially strengthen results due to R. Turner. We obtain information about the operator realizing the similarity transformation, present estimates for the eigenvalues and eigenvectors of the perturbed operator, and also study the inverse problem of spectral analysis.     

Direct and inverse results in Hölder norms

We present a general approach to obtain direct and inverse results for approximation in Hölder norms. This approach is used to obtain a collection of new results related with estimates of the best polynomial approximation and with the approximation by linear operators of non-periodic functions in Hölder norms.     

Norm convergence of multiple ergodic averages on amenable groups

We consider abstract non-negative self-adjoint operators on L²(X) which satisfy the finite-speed propagation property for the corresponding wave equation. For such operators, we introduce a restriction type condition, which in the case of the standard Laplace operator is equivalent to (p, 2) restriction estimate of Stein and Tomas. Next, we show that in the considered abstract setting, our restriction type condition implies sharp spectral multipliers and endpoint estimates for the Bochner-Riesz summability. We also observe that this restriction estimate holds for operators satisfying dispersive or Strichartz estimates. We obtain new spectral multiplier results for several second order differential operators and recover some known results. Our examples include Schrödinger operators with inverse square potentials on Rⁿ, the harmonic oscillator, elliptic operators on compact manifolds, and Schr¨odinger operators on asymptotically conic manifolds.     

Perturbation Analysis of Moore–Penrose Quasi-linear Projection Generalized Inverse of Closed Linear Operators in Banach Spaces

In this paper, we investigate the perturbation problem for the Moore–Penrose bounded quasi-linear projection generalized inverses of a closed linear operaters in Banach space. By the method of the perturbation analysis of bounded quasi-linear operators, we obtain an explicit perturbation theorem and error estimates for the Moore–Penrose bounded quasi-linear generalized inverse of closed linear operator under the T-bounded perturbation, which not only extend some known results on the perturbation of the oblique projection generalized inverse of closed linear operators, but also extend some known results on the perturbation of the Moore–Penrose metric generalized inverse of bounded linear operators in Banach spaces.     

Positive Invertibility of Nonselfadjoint Operators

The paper deals with a class of nonselfadjoint operators in a separable Hilbert lattice. Conditions for the positive invertibility are derived. Moreover, upper and lower estimates for the inverse operator are established. In addition, bounds for the positive spectrum are suggested. Applications to integral operators, integro-differential operators and infinite matrices are discussed.     

On power-bounded operators and operators satisfying a resolvent condition

We obtain, by means of a classification of the eigenvalues, local estimates for holomorphic. functions of a class of linear operators on a finite dimensional linear vector space. We apply these methods to find new proofs of some theorems ofKreiss andMorton, and in addition we give a local estimate of the powers of the inverse of any nonsingular operator in this class.     

Carleman estimate for a strongly damped wave equation and applications to an inverse problem

In this paper, we establish a Carleman estimate for a strongly damped wave equation in order to solve a coefficient inverse problems of retrieving a stationary potential from a single time‐dependent Neumann boundary measurement on a suitable part of the boundary. This coefficient inverse problem is for a strongly damped wave equation. We prove the uniqueness and the local stability results for this inverse problem. The proof of the results relies on Carleman estimate and a certain energy estimates for hyperbolic equation with strongly damped term. Moreover, this method could be used for a similar inverse problem for an integro‐differential equation with hyperbolic memory kernel. Copyright © 2012 John Wiley & Sons, Ltd.     

Perturbation analysis for oblique projection generalized inverses of closed linear operators in Banach spaces

In this paper, we study the perturbation problem for oblique projection generalized inverses of closed linear operators in Banach spaces. By the method of the perturbation analysis of linear operators, we obtain an explicit perturbation theorem and error estimates for the oblique projection generalized inverse of closed linear operators under the T-bounded perturbation, which extend the known results on the perturbation of the oblique projection generalized inverse of bounded linear operators in Banach spaces.     

On a Sequence of Positive Linear Operators Associated with a Continuous Selection of Borel Measures

In this paper we introduce and study a new sequence of positive linear operators acting on the space of Lebesgue-integrable functions on the unit interval. These operators are defined by means of continuous selections of Borel measures and generalize the Kantorovich operators. We investigate their approximation properties by presenting several estimates of the rate of convergence by means of suitable moduli of smoothness. Some shape preserving properties are also shown. Dedicated to the memory of Professor Aldo Cossu     

Carleman estimates for the regularization of ill-posed Cauchy problems

This work is a survey of results for ill-posed Cauchy problems for PDEs of the author with co-authors starting from 1991. A universal method of the regularization of these problems is presented here. Even though the idea of this method was previously discussed for specific problems, a universal approach of this paper was not discussed, at least in detail. This approach consists in constructing of such Tikhonov functionals which are generated by unbounded linear operators of those PDEs. The approach is quite general one, since it is applicable to all PDE operators for which Carleman estimates are valid. Three main types of operators of the second order are among them: elliptic, parabolic and hyperbolic ones. The key idea is that convergence rates of minimizers are established using Carleman estimates. Generalizations to nonlinear inverse problems, such as problems of reconstructions of obstacles and coefficient inverse problems are also feasible.     

On the a posteriori estimates for inverse operators of linear parabolic equations with applications to the numerical enclosure of solutions for nonlinear problems

We consider the guaranteed a posteriori estimates for the inverse parabolic operators with homogeneous initial-boundary conditions. Our estimation technique uses a full-discrete numerical scheme, which is based on the Galerkin method with an interpolation in time by using the fundamental solution for semidiscretization in space. In our technique, the constructive a priori error estimates for a full discretization of solutions for the heat equation play an essential role. Combining these estimates with an argument for the discretized inverse operator and a contraction property of the Newton-type formulation, we derive an a posteriori estimate of the norm for the infinite-dimensional operator. In numerical examples, we show that the proposed method should be more efficient than the existing method. Moreover, as an application, we give some prototype results for numerical verification of solutions of nonlinear parabolic problems, which confirm the actual usefulness of our technique.     

A Sum Operator with Applications to Self-Improving Properties of Poincaré Inequalities in Metric Spaces

We define a class of summation operators with applications to the self-improving nature of Poincaré–Sobolev estimates, in fairly general quasimetric spaces of homogeneous type. We show that these sum operators play the familiar role of integral operators of potential type (e.g., Riesz fractional integrals) in deriving Poincaré–Sobolev estimates in cases when representations of functions by such integral operators are not readily available. In particular, we derive norm estimates for sum operators and use these estimates to obtain improved Poincaré–Sobolev results.     

The Genuine Bernstein–Durrmeyer Operators Revisited

We present some direct and inverse results for the approximation by the genuine Bernstein–Durrmeyer operators U _n . We also consider iterates of U _n and discuss some convergence results towards the corresponding semigroup. Using the eigenstructure of the operators U _n we give new proofs of some known qualitative results and obtain new quantitative estimates concerning the convergence of iterates towards the semigroup.     

An approximation theory for the identification of nonlinear volterra equations

An abstract approximation framework and convergence theory for Galerkin approximations to inverse problems involving nonlinear Volterra integral equations is developed. The approach relies on the theory of m-accretive operators in Banach spaces. An application to heat flow in materials with memory is also discussed.