Strongly P-Positive Operators and the Cayley Transform: Theory and Applications

We present a unifying framework with strongly positive, strongly P-positive operators and the Cayley transform in Banach spaces in order to construct explicit representations of solutions of operator (in particular, differential operator) equations. Such representations are the basis for various new approximations with high (spectral) accuracy. The new concept of P-positive operators appears to be crucial in the effective treatment of the second order differential operator equations and their numerical approximations.

We show how classical evolutionary partial differential equations (e.g. the wave and the Poisson-Darboux equations as well as equations describing linear viscous flows), evolutionary equations with singular integral operator coefficients (describing linear sloshing) as well as Lyapunov and Silvester equations can be treated as special cases of our general theory.