Operator calculus

Philip Joel Feinsilver

Vol. 78 (1978), No. 1, 95–116

DOI: 10.2140/pjm.1978.78.95

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) = e^zC1 where C = e^tLxe^−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 u_n = Cⁿf satisfy the evolution equation u_t = Lu and the eigenvalue equations Au_n = nu_n, thus generalizing the powers xⁿ. We consider processes on R^N as well as R¹ and discuss various examples and extensions of the theory.

Mathematical Subject Classification 2000

Primary: 60J65

Secondary: 58G32, 82A05, 47D05

Milestones

Received: 16 November 1976

Published: 1 September 1978