Operator calculus

Abstract

FEINSILVER To an analytic function L(z) we associate the differential operator L(D), D denoting differentiation with respect to a real variable x. We interpret L as the generator of a process with independent increments having exponential martingale m(x(t), t)= exp (zx(t) -tL{z)). Observing that m(x, -t)-e zC l where C-e tL xe~t L , we study the operator calculus for C and an associated generalization of the operator xD, A-CD. We find what functions / have the property that u n ~Cn f satisfy the evolution equation u t ~Lu and the eigenvalue equations Au n -nu n , thus generalizing the powers x n . We consider processes on R N as well as R 1 and discuss various examples and extensions of the theory.