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
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