Noncommutative Mathematics for Quantum Systems

68 Noncommutative Mathematics for Quantum Systems

GX+Y(z) =〈Ω,(z−X−Y)−^1 Ω〉=

〈

1 ,(z−MxP 2 −My)−^11

〉

=

〈

1 ,

1

z−y

+

xGν(z)

(z−y)( 1 −xGν(z))

〉

=

∫

R×R

1

(z−y)( 1 −xGν(z))

dμ⊗ν

=

∫

R

Gν(z)

1 −xGν(z)

dμ(x) =Gμ

(

1

Gν(z)

)

=Gμ

(

Fν(z)

)

,

or

FX+Y(z) =

1

GX+Y(z)

=

1

Gμ

(

Fν(z)

)=Fμ

(

Fν(z)

)

=Fμ.ν(z).

Multiplicative monotone convolution onM 1 (R+)

Definition 1.7.19 [Ber05a] Let μ and ν be two probability

measures on the positive half-lineR+with transformsKμandKν.

Then themultiplicative monotone convolutionofμandνis defined as

the unique probability measureλ=μmνonR+with transform

Kλ=Kμ◦Kν.

It follows from Subsection 1.7.1 that the multiplicative monotone

convolution onM 1 (R+)is well defined.

Let us first recall some basic properties of the multiplicative

monotone convolution.

Proposition 1.7.20 The multiplicative monotone convolution

M 1 (R+)is associative and∗-weakly continuous in both arguments. It

is affine in the first argument and convolution from the right by a Dirac

measure corresponds to dilation, that is,μmδα =D−α^1 μforα∈R+,

where Dα:R+→R+is defined by Dα(t) =αt.

This convolution is not commutative, that is, in generalμmν 6 =

νmμ. As in the additive case, it is not affine in the second argument,

either, and convolution from the left by a Dirac mass is in general

not equal to a dilation.

We want to extend [Fra06a, Corollary 4.3] to unbounded

positive operators, that is, we want to show that ifXandYare two

positive operators such that X− 1 and Y are monotonically

independent, then the distribution of

√

XY

√

X is equal to the

multiplicative monotone convolution of the distributions ofXand

Y. By Theorem 1.7.13, it is sufficient to do the calculations for the

case whereXandYare constructed from multiplication with the

coordinate functions and the projectionP 2.