Vol. 78, No. 1, 1978

Recent Issues
Vol. 329: 1
Vol. 328: 1  2
Vol. 327: 1  2
Vol. 326: 1  2
Vol. 325: 1  2
Vol. 324: 1  2
Vol. 323: 1  2
Vol. 322: 1  2
Online Archive
The Journal
About the journal
Ethics and policies
Peer-review process
Submission guidelines
Submission form
Editorial board
ISSN: 1945-5844 (e-only)
ISSN: 0030-8730 (print)
Special Issues
Author index
To appear
Other MSP journals
Operator calculus

Philip Joel Feinsilver

Vol. 78 (1978), No. 1, 95–116

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) = ezC1 where C = etLxetL, 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.

Mathematical Subject Classification 2000
Primary: 60J65
Secondary: 58G32, 82A05, 47D05
Received: 16 November 1976
Published: 1 September 1978
Philip Joel Feinsilver
Southern Illinois University
United States