Operator calculus

Abstract

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)-ezCl where C=etLxe−tL, we study the operator calculus for C and an associated generalization of the operator xD, A=CD. We find what functions f have the property that un=Cnf satisfy the evolution equation ut=Lu and the eigenvalue equations Aun=nun, thus generalizing the powers xn. We consider processes on RN as well as R1 and discuss various examples and extensions of the theory. © 1978 by Pacific Journal of Mathematics.