Independence and L ́evy Processes in Quantum Probability 69
Proposition 1.7.21 Letμandνbe two probability measures onR+,
ν 6 =δ 0 , and let Mybe the self-adjoint operator on L^2 (R+×R+,μ⊗ν)
defined by multiplication with the coordinate function(x,y)7→y. Define
Sxon L^2 (R+×R+,μ⊗ν)by
(Sxψ)(x,y) = (x− 1 )
∫
R+
ψ(x,y)dν(y) +ψ(x,y) (1.7.5)
on
DomSx=
{
ψ∈L^2 (R+×R+,μ⊗ν);
∫
R+
xψ(x,y)dν(y)∈L^2 (R+,μ)
}
.
Then Sx− 1 and Myare monotonically independent w.r.t. to the constant
function, and the operator z−
√
SxMy
√
Sxhas a bounded inverse for all
z∈C\R, given by
(
(z−
√
SxMy
√
Sx)−^1 ψ
)
(x,y) =
ψ(x,y) +g(x)
z−y
+h(x).
(1.7.6)
where
g(x) =
√
x−x
( 1 −x)zGν(z) +x
∫
R+
ψ(x,y)dν(y)
+
z(x− 1 )
( 1 −x)zGν(z) +x
∫
R+
ψ(x,y)
z−y
dν(y),
h(x) =
(
√
x− 1 )^2 Gν(z)
( 1 −x)zGν(z) +x
∫
R+
ψ(x,y)dν(y)
+
√
x−x
( 1 −x)zGν(z) +x
∫
R+
ψ(x,y)
z−y
dν(y).
Proof Fixz∈C+. Letx>0, then
Im
z
z−x
=−
xImz
(Rez−x)^2 + (Imz)^2
<0,
and therefore
ImzGν(z) =Im
∫
R+
z
z−x
dν(x)<0.