Noncommutative Mathematics for Quantum Systems

Independence and L ́evy Processes in Quantum Probability 79

Proposition 1.7.35 Letμandνbe two probabilities onRand define
operators Nxand Nyas in Proposition 1.7.31. Then Nxand Nyare self-

adjoint and boolean-independent w.r.t.ω =





1

0

0



. Furthermore, the

operator z−Nx−Nyhas a bounded inverse for all z∈C\R, given by

(z−Nx−Ny)−^1





α

ψ 1

ψ 2



=







β

ψ 1 +βx−cx

z−x

ψ 2 +βy−cy

z−y





, (1.7.10)

where

β=

αGμ(z)Gν(z) +Gν(z)

∫

R

ψ 1 (x)

z−xdμ(x) +Gμ(z)

∫

R

ψ 2 (y)

z−ydν(y)

Gμ(z) +Gν(z)−zGμ(z)Gν(z)

,

(1.7.11)

and cx,cy∈Chave to be chosen such that

∫

R

ψ 1 (x) +βx−cx

z−x

dμ(x) = 0 =

∫

R

ψ 2 (y) +βy−cy

z−y

dν(y).

(1.7.12)

Note that Equation (1.7.12) yields the following formulae for the
constantscx,cy,

cx =

∫ ψ 1 (x)

z−xdμ(x) +β

(

zGμ(z)− 1

)

Gμ(z)

,

cy =

∫ ψ 2 (y)

z−ydν(y) +β

(

zGν(z)− 1

)

Gν(z)

.

Proof NxandNyare boolean-independent by Proposition 1.7.31.
Forz∈C+, we have ImFμ(z)≥Imz>0, ImFν(z)≥Imz>0,
and therefore

Im

Gμ(z) +Gν(z)−zGμ(z)Gν(z)

Gμ(z)Gν(z)

=Im

(

Fμ(z) +Fν(z)−z

)

>0.

This shows that the denominator of the right-hand side of Equation

(1.7.11) cannot vanish forz∈C+. SinceGμ(z) =Gμ(z),Gν(z) =