This agrees with the definition of the spectrum in the matrix case, where the resolvent set comprises all complex numbers that are not eigenvalues. In terms of its spectrum, we will see that a compact operator behaves like a matrix, in the sense that its spectrum is the union of all of its eigenvalues and 0. We begin with the eigenspaces of a compact operator. We start with two lemmas that we w...

This work establishes nonlinear orbital asymptotic stability of scalar radiative shock profiles, namely, traveling wave solutions to the simplified model system of radiating gas [8], consisting of a scalar conservation law coupled with an elliptic equation for the radiation flux. The method is based on the derivation of pointwise Green function bounds and description of the linearized solution ...

We study the asymptotics of the lattice point counting function N(x; y; r) = #f° 2 ¡ : d(x; °y)g for X a Riemannian symmetric space of rank one and ¡ a discontinuous group of motions in X, such that ¡nX has ̄nite volume. We show that

Let T (t), 0 ≤ t < ∞, be a one parameter c0-semigroup of bounded linear operators on a Banach space X with infinitesimal generator A and R(λ, A) be the resolvent operator of A. The Hille-Yosida Theorem for c0-semigroups asserts that the resolvent operator of the infinitesimal generator A satisfies ‖R(λ, A)‖ ≤ M λ−ω for some constants M > 0 and λ ∈ R (the set of real numbers), λ > ω. The object ...

The paper reports on a recent construction ofM -functions and Krĕın resolvent formulas for general closed extensions of an adjoint pair, and their implementation to boundary value problems for second-order strongly elliptic operators on smooth domains. The results are then extended to domains with C Hölder smoothness, by use of a recently developed calculus of pseudodifferential boundary operat...

We consider solutions to the linear wave equation on non-compact Riemannian manifolds without boundary when the geodesic flow admits a filamentary hyperbolic trapped set. We obtain a polynomial rate of local energy decay with exponent depending only on the dimension.

For a strongly elliptic second-order operator A on a bounded domain Ω ⊂ R it has been known for many years how to interpret the general closed L2(Ω)-realizations of A as representing boundary conditions (generally nonlocal), when the domain and coe cients are smooth. The purpose of the present paper is to extend this representation to nonsmooth domains and coe cients, including the case of Höld...

We transform the two-matrix model, studied by P.Di Francesco and al. in [1], into a normal one-matrix model by identifying a " formal " integral used by these authors as a proper integral. We show also, using their method, that the results obtained f or the resolvent and the density are not reliable.

Article history: Received 24 September 2009 Accepted 6 November 2009 Available online 20 February 2010 Submitted by R. Bhatia AMS classification: 15B48 15B36 91A40 91B66