Spectra and dynamics of bounded pseudo-differential operators (Q1382136)

The author studies the spectral properties of the pseudodifferential operator \(a(D)\) with symbol \(a(\xi)\) in the class \(S^0 (\mathbb{R}^n)\). It is known that \(a(D)\) is bounded from \(L^p (\mathbb{R}^n)\) to \(L^p (\mathbb{R}^n)\) for \(1<p <\infty\); the author computes in such a frame the spetrum and the essential spectrum of \(a(D)\). An application is given concerning the asymptotic stability of the zero solution of a related semilinear equation.