GENERALIZED DYSON SERIES AND FEYNMAN'S OPERATIONAL CALCULUS 7

- / 2 N 2 N

D. 3C(L (R )): The space of bounded linear operators from L (R )

into itself.

The notation | | • | | will be used both for the norm of vectors and for

the norm of operators; the meaning will be clear from the context.

E. The semigroup exp(-zHQ): We give some facts which we will use

frequently concerning the holomorphic semigroup {exp(-zHn)}

r

~ generated

U Z £

L

i

by the "free Hamiltonian" HQ - -(1/2)^.132/ax^ in L2(RN). (See [26,

Chap. IX, §1.8, pp. 495-497].) We use notation convenient for our pur-

poses. The operators {exp[-s(HQ/X)]: s 0, X e C^.} are all in 3C(L2(RN))

and satisfy:

(0.6) | [exp[-s(H0/X)]|| _ 1.

In fact, when X e C, is purely imaginary, exp[-s(Hn/x)] is a unitary

° -s(H0/X)

operator. As a function of X, exp[-s(HQ/X)], also denoted by e

in this paper, is analytic in C, and continuous in the strong operator

topology (or strongly continuous) in C~ (Recall that for operator-

valued (or for vector-valued) functions, all the natural notions of

analyticity coincide. See [15, §3.10, esp. Theorem 3.10.1, p. 93].)

Next, we state a familiar explicit formula for the operator exp[-s(H0/X)].

Given if i e L2(RN),

o

(0.7) (exp[-s(H0/X)]*)(O = (277)N/2 / *(u) exp[-Xil2u;^l ]du.

RN

The integral in (0.7) exists as an ordinary Lebesgue integral for

X € C,, but, when X is purely imaginary and ty is not integrable, the

integral should be interpreted in the mean just as in the theory of the

Fourier-Plancherel transform.

As is well known, the (negative) normalized Laplacian HQ is the

generator of the Brownian motion on E : it follows, in particular, that

the semigroup {exp(-sHn)}

0

is intimately connected with Wiener

measure m defined in I below. (See [13, Chap. 3; 14, Chap. 3; and esp.

46, Chap. II].)

F. M(0,t): Let t 0 be fixed. M(0,t) will denote the space of

complex Borel measures n on the open interval (0,t). For information on