Noncommutative Mathematics for Quantum Systems

56 Noncommutative Mathematics for Quantum Systems

and itsreciprocal Cauchy–Stieltjes transform Fμby

Fμ(z) =

1

Gμ(z)

.

Denote byFthe following class of holomorphic self-maps,

F=

{

F:C+→C+;Fholomorphic and inf

z∈C+

ImF(z)

Imz

= 1

}

.

The mapμ 7→Fμdefines a bijection between the classM 1 (R)of
probability measures onRandF, as follows from the following
theorem.

Theorem 1.7.1 [Maa92] Let F:C+→C+be holomorphic, then the
following are equivalent.

(i)infz∈C+ImImF(zz)=1;

(ii) there exists aμ∈M 1 (R)such that F=Fμ.

Furthermore,μis uniquely determined by F.

Similarly, forμa probability measure on the unit circleT={z∈
C;|z|= 1 }or on the positive half-lineR+={x∈R;x≥ 0 }, we
define

ψμ(z) =

∫ xz

1 −xz

dμ

and

Kμ(z) =

ψμ(z)

1 +ψμ(z)

forz∈C\suppμ.
The mapμ7→Kμdefines bijections between the classM 1 (T)of
probability measures onTand the class

S={K:D→D;Kholomorphic andK( 0 ) = 0 },

whereD={z∈C;|z|< 1 }, and between the classM 1 (R+)of
probability measures onR+and the class

P=







K:C\R+→C\R+;

Kholomorphic

limt↗ 0 K(t) =0,K(z) =K(z),

argz≤argK(z)≤πfor allz∈C+







,