Real sectorial operators

Sectorial operators that act in complex Banach spaces and map real subspaces into themselves should be called real sectorial operators. These operators have already been used implicitly in the study of various diffusion equations. Meanwhile, in the Łojasiewicz–Simon theory which provides longtime convergence of solutions to stationary solutions, the real valued Lyapunov functions play an important role. In order to make general methods for studying longtime convergence problems on the basis of the Łojasiewicz–Simon theory, it may therefore be meaningful to give an explicit definition for these real sectorial operators and to show their basic properties that are inherited from those of complex sectorial operators.