Independence and L ́evy Processes in Quantum Probability 81Substituting Equation (1.7.11) into this expression, we get the
desired resultzβ−cx−cy=α.
Theorem 1.7.36 Let X and Y be two self-adjoint operators on a Hilbert
space H that are boolean-independent w.r.t. a unit vectorΩ ∈ H and
assume thatΩis cyclic, that is, that
alg{h(X),h(Y);h∈Cb(R)}Ω=H.Then X+Y is essentially self-adjoint and the distribution w.r.t.Ωof the
closure of X+Y is equal to the boolean convolution of the distributions of
X and Y w.r.t.Ω, that is,
L(X+Y,Ω) =L(X,Ω)]L(Y,Ω).Proof Letμ=L(X,Ω),ν=L(Y,Ω).
By Theorem 1.7.32 and Lemma 1.7.8 it is sufficient to consider
the case whereXandYare defined as in Proposition 1.7.31. Then
Proposition 1.7.35 shows thatz−X−Yadmits a bounded inverse
for allz ∈C\Rand therefore that Ran(z−X−Y)is dense. By
[RS80, Theorem VIII.3] this is equivalent toX+Ybeing essentially
self-adjoint.
Using Equation (1.7.10), we can compute the Cauchy transform
of the distribution of the closure ofX+Y. Letz∈C+, then
GX+Y(z)
= 〈Ω,(z−X−Y)−^1 Ω〉=〈
ω,(z−Nx−Ny)−^1 ω〉=〈
1
0
0, Gμ(z)Gν(z)
Gμ(z) +Gν(z)−zGμ(z)Gν(z)
1x−zGGμμ((zz))−^1
z−x
y−zGGνν((zz))−^1
z−y
〉=Gμ(z)Gν(z)
Gμ(z) +Gν(z)−zGμ(z)Gν(z).Replacing all Cauchy transforms by their reciprocals, this becomes
FX+Y(z) =Fμ(z) +Fν(z)−z=Fμ]ν(z).