Abstract:We shall show that every differential operator of 2-nd order in a real separable Hilbert space can be decomposed into a regular and an irregular operator. Then we shall characterize irregular operators and differential operators satisfying the maximum principle. Results obtained for the L\'evy laplacian in [3] will be generalized for irregular differential operators satisfying the maximum principle.
Keywords: L\'evy laplacian, maximum principle, Dirichlet and Poisson problem
AMS Subject Classification: 31C45, 46C99, 47F05