Towards an L1-theory for vector-valued elliptic boundary value problems
2003, Denk, Robert, Hieber, Matthias, Prüss, Jan
In this note, we consider vector-valued boundary value problems with constant coefficients in the L1-setting for a half space. We assume the Lopatinskii-Shapiro condition to be true and obtain a representation of the solution u of the elliptic problem by integral operators which allows to deduce a-priori estimates for u in the L1-norm. These estimates imply in particular that the L1-realization of an elliptic boundary value problem with constant coefficients in the half space generates an analytic C0-semigroup.
The spectrum of the multiplication operator associated with a family of operators in a Banach space
2006, Denk, Robert, Möller, Manfred, Tretter, Christiane
An operator family of densely defined closed linear operators and the multiplication operator associated with it are considered. The spectrum of this multiplication operator is expressed in terms of the spectra of the operators in the given family.