ON P-DEPENDENT LOCAL SPECTRAL PROPERTIES OF CERTAIN LINEAR-DIFFERENTIAL OPERATORS IN L-P(R-N)

The aim is to investigate certain spectral properties, such as decompo sability, the spectral mapping property and the Lyubich-Matsaev proper ty, for linear differential operators with constant coefficients land more general Fourier multiplier operators) acting in L-p(R-N). The cri teria developed for such operators are quite general and p-dependent, i.e. they hold for a range of p in an interval about 2 (which is typic ally not (1, alpha)). The main idea is to construct appropriate functi onal calculi: this is achieved via a combination of methods from the t heory of Fourier multipliers and local spectral theory.