We characterize the well-posedness for second order discrete evolution equations in UMD spaces by means of Fourier multipliers and R-boundedness properties of the resolvent operator which defines the equation. Applications to semilinear problems are given.

Abstract The spectral theory on the S -spectrum was introduced to give an appropriate mathematical setting quaternionic quantum mechanics, but it soon realized that there were different applications of this theory, for example, fractional heat diffusion and Dirac operator manifolds. In seminal paper we introduce harmonic functional calculus based integral representation axially functions. This ...

We extend the concept of Lifshits–Krein spectral shift function associated with a pair of self-adjoint operators to the case of pairs of (admissible) operators that are similar to self-adjoint operators. An operator H is called admissible if: (i) there is a bounded operator V with a bounded inverse such that H = V −1 HV for some self-adjoint operator H; (ii) the operators H and H are resolvent ...

We describe methods which have been used to analyze the spectrum of non-self-adjoint differential operators, emphasizing the differences from the self-adjoint theory. We find that even in cases in which the eigenfunctions can be determined explicitly, they often do not form a basis; this is closely related to a high degree of instability of the eigenvalues under small perturbations of the opera...

This note discusses some aspects of the analysis leading to the proof of the main theorem in [10] (stated here as Theorem 1) on the structure of the asymptotics of the resolvent trace of a general elliptic cone operator as the spectral parameter tends to infinity, under suitable minimal growth assumptions on the principal symbols of the operator. We deal with an elliptic cone differential operator

We study the relativistic Lee model in static Riemannian manifolds. The model is constructed through its resolvent, which is based on the socalled principal operator and the heat kernel techniques. It is shown that making the principal operator well-defined dictates how to renormalize the parameters of the model. The underlying geometry is found not to affect the ultra-violet behavior of the th...

Modifying von Neumann’s alternating projections algorithm, we obtain an alternating method for solving the recently introduced Common Solutions to Variational Inequalities Problem (CSVIP). For simplicity, we mainly con…ne our attention to the two-set CSVIP, which entails …nding common solutions to two unrelated variational inequalities in Hilbert space. Keywords: Alternating method, averaged op...

For a time-independent Schrödinger equation with Hamiltonian operator H = −∆ + V on L(R ) we call Gz = RzV , the Green function operator, where Rz is the resolvent of −∆. We consider an iteration scheme that is based on the operator Gz. Under standard conditions on the potential V , which include the Coulomb interaction, we prove the convergence of the iteration to the ground state energy and e...