Starting from the classic definitions of resolvent set and spectrum of a linear bounded operator on a Banach space, we introduce the local resolvent set and local spectrum, the local spectral space and the single-valued extension property of a family of linear bounded operators on a Banach space. Keeping the analogy with the classic case, we extend some of the known results from the case of a l...

In this paper, we consider a more general form of variational inclusions, called generalized variational inclusion (for short, GVI). In connection with GVI, we also consider a generalized resolvent equation with H-resolvent operator, called H-resolvent equation (for short, H-RE). We suggest iterative algorithms to compute the approximate solutions of GVI and H-RE. The existence of a unique solu...

We investigate analytic perturbations of the reduced resolvent of a nite-dimensional linear operator (also known as Drazin inverse in the linear algebra literature). Our approach is based on spectral theory of linear operators as well as on a new notion of group reduced resolvent. It allows to treat regular and singular perturbations in a uniied framework. We provide an algorithm for computing ...

We give a new characterization of the solvability of an abstract Cauchy problems in terms of moment sequences, using the resolvent operator at only one point.

Sensitivity analysis for relaxed cocoercive variational inclusions based on the generalized resolvent operator technique is discussed The obtained results are general in nature.

Finite-rank expansions of the two-body resolvent operator are explored as a tool for calculating the full three-dimensional two-body T -matrix without invoking the partial-wave decomposition. The separable expansions of the full resolvent that follow from finite-rank approximations of the free resolvent are employed in the Low equation to calculate the T-matrix elements. The finite-rank expansi...

We consider the problem of determining the unknown term in the right-hand side of a second-order differential equation with unbounded operator generating a cosine operator function from the overspecified boundary data. We obtain necessary and sufficient conditions of the unique solvability of this problem in terms of location of the spectrum of the unbounded operator and properties of its resol...

For a class of manifolds X that includes quotients of real hyperbolic (n + 1)-dimensional space by a convex co-compact discrete group, we show that the resonances of the meromorphically continued resolvent kernel for the Laplacian on X coincide, with multiplicities, with the poles of the meromorphically continued scattering operator for X. In order to carry out the proof, we use Shmuel Agmon’s ...

We define the complex powers of a densely defined operator A whose resolvent exists in a suitable region of the complex plane. Generally, this region is strictly contained in an angle and there exists α ∈ [0,∞) such that the resolvent of A is bounded by O((1 + |λ|)α) there. We prove that for some particular choices of a fractional number b, the negative of the fractional power (−A)b is the c.i....