We provide a geometrical interpretation for the best approximation of the discrete harmonic oscillator equation formulated in a general Banach space setting. We give a representation of the solution, and a characterization of maximal regularity-or well posednesssolely in terms of R-boundedness properties of the resolvent operator involved in the equation.

We consider scattering by short range perturbations of the semi-classical Laplacian. We prove that when a polynomial bound on the resolvent holds the scattering amplitude is a semi-classical Fourier integral operator associated to the scattering relation near a non-trapped ray. Compared to previous work, we allow the scattering relation to have more general structure.

We present a new method for computation of the Korteweg–de Vries hierarchy via heat invariants of the 1-dimensional Schrödinger operator. As a result new explicit formulas for the KdV hierarchy are obtained. Our method is based on an asymptotic expansion of resolvent kernels of elliptic operators due to S. Agmon and Y. Kannai.

In this manuscript, we have a tendency to execute Banach contraction fixed point theorem combined with resolvent operator to analyze the exact controllability results for fractional neutral integrodifferential systems (FNIDS) with state-dependent delay (SDD) in Banach spaces. An illustration is additionally offered to exhibit the achieved hypotheses.

It is shown that, for the spectral operators of scalar type, the well-known characterizations of the generation of C0and analytic semigroups of bounded linear operators can be reformulated exclusively in terms of the spectrum of such operators, the conditions on the resolvent of the generator being automatically met and the corresponding semigroup being that of the exponentials of the operator.

The aim of this work is to use resolvent operator technique to find the common solutions for a system of generalized nonlinear relaxed cocoercive mixed variational inequalities and fixed point problems for Lipschitz mappings in Hilbert spaces. The results obtained in this work may be viewed as an extension, refinement and improvement of the previously known results.

In this paper we consider a compact manifold with boundary X equipped with a scattering metric g as defined by Melrose [9]. That is, g is a Riemannian metric in the interior of X that can be brought to the form g = x dx + xh near the boundary, where x is a boundary defining function and h is a smooth symmetric 2-cotensor which restricts to a metric h on ∂X. Let H = ∆ + V where V ∈ xC(X) is real...

For a bounded linear operator A on a Banach space we characterize the isolated points in the spectrum of A , the Riesz points of A , and the poles of the resolvent of A . 1. Terminology and introduction Throughout this paper E will be an infinite-dimensional complex Banach space and A will be a bounded linear operator on E. We denote by N(A) the kernel and by A(E) the range of A. The spectrum o...

In this paper we consider the Maxwell resolvent operator and its finite element approximation. In this framework it is natural the use of the edge element spaces and to impose the divergence constraint in a weak sense with the introduction of a Lagrange multiplier, following an idea by Kikuchi [14]. We shall review some of the known properties for edge element approximations and prove some new ...

The operator e−tA and the heat trace Tr e−tA, for t > 0, are investigated in the case when A is an elliptic differential operator on a manifold with conical singularities. Under a certain spectral condition (parameter–ellipticity) we obtain a full asymptotic expansion in t of the heat trace as t → 0+. As in the smooth compact case, the problem is reduced to the investigation of the resolvent (A...