Vol. 78, No. 1, 1978

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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