A theory of generalized functions based on holomorphic semi-groups Part A: introduction and survey

AbstractIn a Hilbert space X consider the evolution equation du dudt=−Au with A a nonnegative unbounded self-adjoint operator. A is the infinitesimal generator of a holomorphic semi-group. Solutions u(· ): (0, ∞) X of this equation are called trajectories. Such a trajectory may or may not correspond to an “initial condition at t=0″ in X. The set of trajectories is considered as a space of generalized functions. The test function space is defined to be SX, A= ⊂t>0 e−, A(X).For the spaces SX, A, TX, A I discuss a pairing, topologies, morphisms, tensor products and kernel theorems. Examples are given