Noncommutative Mathematics for Quantum Systems

60 Noncommutative Mathematics for Quantum Systems

alg{h(X),h(Y);h∈Cb(R)}Ω=H.

Then the products

√

XY

√

X and

√

YX

√

Y are essentially

self-adjoint and positive, and their distributions w.r.t. toΩare

equal to the free convolution of the distributions of X and Y w.r.t.

Ω, that is,

L(

√

XY

√

X,Ω) =L(

√

YX

√

Y,Ω) =L(X,Ω)L(Y,Ω).

1.7.3 A useful Lemma

To deal with unitary equivalence of possibly unbounded normal
operators, we will use several times the following lemma.

Lemma 1.7.8 Let X and X′be possibly unbounded normal operators
on Hilbert spaces H and H′. Assume that there exists a unitary operator
U:H→H′such that

U f(X) =f(X′)U (1.7.1)

for any bounded continuous function f onC.
Then UX=X′U.

To prove this lemma, one can first prove the analogous statement
for self-adjoint operators, e.g., using resolvents, and then apply the
properties of the decompositionX= A+iBof normal operators
as a linear combination of two commuting self-adjoint operators,
cf.[Ped89, Proposition 5.1.10]. It is actually sufficient to require
Condition (1.7.1) for a much smaller class of functions, for
example, compactly supported real-valuedC∞-functions.

1.7.4 Monotone convolutions

Definition 1.7.9 [Mur00] LetA 1 ,A 2 ⊂B(H)be two∗-algebras of
bounded operators on a Hilbert spaceH, and letΩ∈Hbe a unit
vector. We say thatA 1 andA 2 aremonotonically independentw.r.t.Ω,
if we have

〈Ω,X 1 X 2 ···XkΩ〉=

〈

Ω, ∏

κ:εκ= 1

XκΩ

〉

∏

κ:εκ= 2

〈Ω,XκΩ〉

for allk∈N,ε∈Ak,X 1 ∈Aε 1 ,... ,Xk∈Aεk.