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We prove that the resolvent of a linear operator pencil is analytic on an open annulus if and only if the coefficients of the Laurent series satisfy a system of fundamental equations and are geometrically bounded. Our analysis extends earlier work on the fundamental equations to include the case where the resolvent has an isolated essential singularity. We find a closed form for the resolvent and use the fundamental equations to establish key spectral separation properties when the resolvent has only a finite number of isolated singularities. Finally we show that our results can also be applied to polynomial pencils., 18 pages: correction to published article

While neural networks have been demonstrated to be highly successful in mathematical and statistical modelling of a comprehensive selection of problems, their application to inverse problems is not without complications. Recent works have shown that neural networks are especially prone to stability issues -- both in a classical sense, and in the context of so called \emph{adverserial attacks} which has come to be regarded as the most pervasive source of instability in modern neural network models. Concomitantly, methods of constructing \emph{invertible neural networks} with diffeomorphic layer structures with \emph{normalizing flows} have been proposed as an interesting method for approaching inverse problems by probabilistic augmentation of latent variable models to induce full-rank in a conditional setting. However, these models can often be prohibitively expensive in terms of computational efficiency and memory usage while displaying sufficiently different architectures as to not be trivially extendable to tools for commonly defined feed-forward neural networks. In this thesis, we motivate the theory of inverse problems via integral equations and spectral theory and discuss the connection of statistical learning theory to neural networks with special focus on encoder-decoder models. Furthermore, in the context of neural networks, we will discuss the underlying theory of epistemic and aleatoric uncertainty, discuss the role of probabilistic modelling, and evaluate the idea of latent probabilistic completion as a remedial method for undercomplete modelling tasks. Our contribution to the subject of invertible neural networks can be summarized as follows. We propose a relatively simple architectural modification of existing encoder-decoder models using both implicit and explicit orthogonal constraints using Riemannian manifold learning and resolvent operators to construct a wider class of invertible neural networks which are compatible with classic feed-forward architectures. We show that these models provide both a significant decrease in the parameter space compared to standard encoder-decoder networks, as well as theoretical guarantees of robustness and stability without significant loss to model performance. We apply these architectures in combination with existing variational Bayesian methods for a generative approach to underdetermined inverse problems. To this end, we introduce a class of piecewise diffeomorphic activation functions and a bijective Gaussian to Dirichlet transformation for latent variables as an alternative to the canonical Softmax transformation, and propose the application of simple conditional additive coupling layers to improve conditioning in generative models.

We extend and generalize some results in local spectral theory for upper triangular operator matrices to upper triangular operator matrices with unbounded entries. Furthermore, we investigate the boundedness of the local resolvent function for operator matrices.

In this work, we give sufficient conditions for the existence of a mild solution for some impulsive integro-differential equations in Banach spaces. We study the existence without assuming the Lipschitz condition on the nonlinear term f. The compactness on the C0-semigroup(T(t))t≥0 in a Banach space is not needed. We use Hausdorff’s measure of noncompactness, resolvent operators and Darbo’s fixed point Theorem to obtain the main result of this work. Keywords: Impulsive integro-differential equations, Mild solutions, Sadovskii’s fixed point Theorem, Resolvent operators, Noncompactness measures

In this work, we consider a class of partially observed stochastic integrodifferential equations on Hilbert spaces subject to measurement uncertainty. We prove the existence of optimal feedback control law from a class of operator valued functions furnished with the product topology. This work is an extension of [2] for uncertain systems governed by stochastic differential equations on Hilbert spaces ; whereby the measurement uncertainty is a square integrable stochastic process. We illustrate the abstract results proved by analyzing an example.

This work concerns the study of the controllability for some nonlinear partial functional integrodifferential equation with finite delay arising in the modelling of materials with memory in Banach spaces. We give sufficient conditions that ensure the controllability of the system by supposing that its undelayed part admits a resolvent operator in the sense of Grimmer, and by making use of the measure of noncompactness and the Monch fixed-point Theorem. As a result, we obtain a generalization of several important results in the literature, without assuming the compactness of the resolvent operator. An example of applications is given for illustration.

In this paper, we study the split common null point problem. Then, using the hybrid projection method and the metric resolvent of monotone operators, we prove a strong convergence theorem for an iterative method for finding a solution of this problem in Banach spaces.

The present paper aims at announcing a new approach to the construction of the asymptotics (at infinity in configuration space) of the resolvent kernel of the Schrodinger operator in the scattering problem of three one-dimensional quantum particles interacting by compactly supported pair repulsive potentials. Within the framework of this approach, the asymptotics of eigenfunctions of the absolutely continuous spectrum of the Schrodinger operator can be constructed explicitly. It should be emphasized that the restriction of the consideration to the case of compactly supported pair potentials does not lead to a simplification of the problem in its essence, since the potential of the interaction of all three particles remains nondecreasing at infinity, but allows one to put aside a certain number of technical details.

We prove $L^p\to L^{p'}$ bounds for the resolvent of the Laplace-Beltrami operator on a compact Riemannian manifold of dimension $n$ in the endpoint case $p=2(n+1)/(n+3)$. It has the same behavior with respect to the spectral parameter $z$ as its Euclidean analogue, due to Kenig-Ruiz-Sogge, provided a parabolic neighborhood of the positive half-line is removed. This is region is optimal, for instance, in the case of a sphere., 14 pages

We dress atoms with multiple-radiofrequency fields and investigate the spectrum of transitions driven by an additional probe field. A complete theoretical description of this rich spectrum is presented, in which we find allowed transitions and determine their amplitudes using the resolvent formalism. Experimentally, we observe transitions up to sixth order in the probe field using radiofrequency spectroscopy of Bose-Einstein condensates trapped in single- and multiple-radiofrequency-dressed potentials. We find excellent agreement between theory and experiment, including the prediction and verification of previously unobserved transitions, even in the single-radiofrequency case., Comment: 20 pages, 7 figures

In this work, we consider a nonlinear resolvent integro-differential evolution inclusions in Hilbert spaces. This paper deals with the approximate controllability for nonlinear resolvent integro-di...

Under a compactness assumption on the resolvent, some properties on relevant operators generated by resolvent are given. Existence results of fractional control systems are obtained by Schauder?s fixed point theorem and approximation techniques. Furthermore, the approximately controllable result is acquired under the assumption that the corresponding linear system is approximately controllable, which improves and extends some results on this topic.

We calculate in a numerically friendly way the Fourier transform of a non-integrable function, such as , by replacing F with R-1FR, where R represents the resolvent for harmonic oscillator Hamiltonian. As contrasted with the non-analyticity of at in the case of a simple replacement of F by , where and represent the momentum and position operators, respectively, the turns out to be an entire function. In calculating the resolvent kernel, the sampling theorem is of great use. The resolvent based Fourier transform can be made supersymmetric (SUSY), which not only makes manifest the usefulness of the even-odd decomposition ofin a more natural way, but also leads to a natural definition of SUSY Fourier transform through the commutativity with the SUSY resolvent.

In this work, we study the existence of mild solutions for some partial functional integrodifferential equations with finite delay in a Frechet spaces. We assume that the linear part has a resolvent operator in the sense given by Grimmer (Trans Am Math Soc 273: 333–349, 1982). The nonlinear part is a sum of a Lipschitzian function and another satisfies the Caratheodory’s conditions. Our approach is based on a nonlinear alternative of Avramescu type and the resolvent operators theory. An application is provided to a reaction-diffusion equation with delay.

In this paper, we introduce a new concept of stability of resolvent operator families, almost exponential stability. We establish a perturbation theorem on analytic resolvent operator families, and derive some sufficient conditions on the almost exponential stability or exponential stability of resolvent operator families by using rescaling technique and contour integral approach. Our theorems generalize corresponding results for semigroups.

The resolvent operator approach of [1] is applied to solve a set-valued variational inclusion problem in ordered Hilbert spaces. The resolvent operator under consideration is called relaxed resolvent operator and we demonstrate some of its properties. To obtain the solution of a set-valued variational inclusion problem, an iterative algorithm is developed and weak-RRD set-valued mapping is used. The problem as well as main result of this paper are more general than many previous problems and results available in the literature.

International audience; We consider a Laplace problem with Dirichlet boundary condition in a three dimensional domain containing an inclusion taking the form of a thin tube with small thickness. We prove convergence in operator norm of the resolvent of this problem as the thickness goes to 0, establishing that the perturbation on the resolvent induced by the inclusion is not greater than some (negative) power of the logarithm of the thickness. We deduce convergence of the eigenvalues of the perturbed operator toward the limit operator.

We construct a semiclassical parametrix for the resolvent of the Laplacian acting on functions on nontrapping conformally compact manifolds with variable sectional curvature at infinity. We apply this parametrix to analyze the Schwartz kernel of the semiclassical resolvent and Poisson operator and to show that the semiclassical scattering matrix is a semiclassical Fourier Integral Operator of appropriate class that quantizes the scattering relation. We also obtain high energy estimates for the resolvent and show existence of resonance free strips of arbitrary height away from the imaginary axis. We then use the results of Datchev and Vasy on gluing semiclassical resolvent estimates to obtain semiclassical resolvent estimates on certain conformally compact manifolds with hyperbolic trapping.

The purpose of this paper is to introduce a new iterative method that is the combination of the proximal point algorithm, viscosity approximation method and alternating resolvent method for finding the common zeros of two accretive operators in Banach spaces. And we will prove the strong convergence theorems for the iterative algorithms and give the example of the main theorems. The results of this paper are improvements and extensions of the corresponding ones announced by many others.

For operators generated by a certain class of infinite three-diagonal matrices with matrix elements we establish a characterization of the resolvent set in terms of polynomial solutions of the underlying second order finite-difference equations. This enables us to describe some asymptotic behavior of the corresponding systems of vector orthogonal polynomials on the resolvent set. We also find that the operators generated by infinite Jacobi matrices have the largest resolvent set in this class.

We use the method of similar operators to perform the asymptotic analysis of the spectrum of a differential operator with an involution. We show that such operators have compact resolvent, and that their large eigenvalues are determined by the values of (the Fourier coefficients) of their potential up to a summable sequence.

We consider weakly nonlinear Fredholm integral equations with degenerate kernel in Banach spaces and establish a necessary condition and sufficient conditions for the existence of solutions of equations of this kind. The convergent iterative procedures are proposed for the construction either of a single solution or of at least one possible solution.

In this paper, we consider the generalized Poisson’s equation associated with Heckman–Opdam Laplacian operator. We introduce and we study the generalized resolvent function associated to the equation. Next, we characterize the spectrum of the hypergeometric Fourier transform via the generalized resolvent function.

In view of ideas for semigroups, fractional calculus, resolvent operator and Banach contraction principle, this manuscript is generally included with existence and controllability (EaC) results for fractional neutral integro-differential systems (FNIDS) with statedependent delay (SDD) in Banach spaces. Finally, an examples are also provided to illustrate the theoretical results.

A divergent-type elliptic operator of arbitrary even order 2m is studied. Coefficients of the operator are -periodic, is a small parameter. The resolvent equation is solvable in the Sobolev space of order m for any , provided the parameter is sufficiently large, , where the bound depends only on constants from ellipticity condition. The limit equation is of the same type but with constant coefficients, that is, . The limit operator can be considered here, for instance, in the sense of G-convergence. We prove that the resolvent approximates in operator -norm with the estimate , as . We find also the approximation of the resolvent in operator -norm. This is the sum , where is a correcting operator whose structure is given. We prove the estimate , as .

We study and characterize the compactness of resolvent families of operators associated to fractional differential equations. We show an application in the study of existence of mild solutions for a class of semilinear fractional differential equations with non-local initial conditions.

It is known that a closed linear operator A, defined densely on a Banach space X, is non-negative if and only if −A generates a bounded analytic γ-times resolvent family for some 0 γ 2 . In this paper, by using such resolvent families as well as the Komatsu representations, we characterize systematically the domains of fractional powers of non-negative operators on Banach spaces. Gaps between the subset ⋃ α > 0 D ( A α ) and the whole space X are discussed as well.

In this work we investigate the resolvent operator and completeness of eigenfunctions of a Sturm–Liouville problem with discontinuities at two points. The problem contains an eigenparameter in one of the boundary conditions. For operator-theoretic formulation of the considered problem we define an equivalent inner product in the Hilbert space \( L_{2} [ - 1,1] \oplus C \) and suitable self-adjoint linear operator in it.

This is the first in a series of papers in which we investigate the resolvent and spectral measure on non-trapping asymptotically hyperbolic manifolds with applications to the restriction theorem, spectral multiplier results and Strichartz estimates. In this first paper, we use semiclassical Lagrangian distributions and semiclassical intersecting Lagrangian distributions, along with Mazzeo-Melrose 0-calculus, to construct the high energy resolvent on general non- trapping asymptotically hyperbolic manifolds, generalizing the work due to Melrose, Sa Barreto and Vasy. We note that there is an independent work by Y. Wang which also constructs the high-energy resolvent., Comment: 49 pages

Some growth conditions of the resolvent function of a Banach space operator are investigated using higher order Cesaro means. More precisely, Abel and Nevanlinna estimates are obtained under the condition of boundedness of some weighted Cesaro averages. Also, certains results related to the (strong or uniform) convergence of Cesaro means are mentioned.

In this manuscript, we investigate the solvability and optimal controls for impulsive fractional stochastic integro-differential equations in Hilbert space. Sufficient conditions are obtained for the existence of mild solution of the considered system by using analytic resolvent operators, the uniform continuity of the resolvent, and Leray---Schauder fixed point theorem. Then, the existence of optimal controls is discussed for the considered system. Finally, the obtained theoretical result is validated through an example.

LetD(A)be the domain of anm-accretive operatorAon a Banach spaceE. We provide sufficient conditions for the closure ofD(A)to be convex and forD(A)to coincide withEitself. Several related results and pertinent examples are also included.

We establish conditions required for the generalized invertibility of Fredholm integral operators with degenerate kernels in Banach spaces. New results on the generalized inversion of Fredholm integral operators with degenerate kernels in Banach spaces are obtained. A detailed analysis of an example is also presented.

A new system of general nonlinear variational inclusions is introduced and studied in Banach spaces. An iterative algorithm is developed and analyzed by use of the resolvent operator techniques to find the approximate solutions of the system of general nonlinear variational inclusions involving different nonlinear operators in uniformly smooth Banach spaces.

We investigate the structure of the essential spectrum of a self-adjoint Fredholm type partial integral operator H. We obtain an explicit description of the essential spectrum of H and prove that an eigenvalue of H exists.

We consider the operatorwhere are matrices , means the tensor product. Norm estimates for the resolvent of that operator are derived. By these estimates, we obtain bounds for a solution of the equationand explore perturbations of that equation. The norm estimates for the resolvent of enable us to establish a bound for the distance between invariant subspaces of two matrices. Besides, the well-known Davis–Kahan result is particularly generalized. In addition, we derive a new stability test for non-linear non-autonomous ordinary differential equations.

In this paper we find a new condition on a real periodic potential for which the self-adjoint Schrodinger operator may be defined by a quadratic form and the spectrum of the operator is purely absolutely continuous. This is based on resolvent estimates and spectral projection estimates in weighted \(L^2\) spaces on the torus, and an oscillatory integral theorem.

An approach describing the spaces of iterations of resolvents with constant mappings is given. Its use allows one to construct (homogeneous, not algebraically homogeneous) cut-point spaces of arbitrary order.

We find the lower boundary for the essential spectrum of a Fredholm type partial integral operator H. We also obtain an estimate for the number of eigenvalues below this boundary.

In this paper, the conditions are found on the coefficients that ensure the existence of the resolvent and the discreteness of spectrum for a class of singular differential operators.