Spectral theory of positive semigroups generated by differential operators (Q1337760)

Let \({\mathcal A}_ p\) be the \(L^ p(\mathbb{R}^ n)\)-realization of a differential operator with constant and real coefficients. Assume that \({\mathcal A}_ p\) generates a positive \(C_ 0\)-semigroup \(T_ p(t)\) on \(L^ p(\mathbb{R}^ n)\). It is shown that then the spectral mapping theorem of the form \[ e^{t\sigma({\mathcal A}_ p)}= \sigma(T_ p(t))\backslash\{0\} \] holds for all \(t\geq 0\).