Noncommutative Mathematics for Quantum Systems

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.