Noncommutative Mathematics for Quantum Systems

70 Noncommutative Mathematics for Quantum Systems

Similarly, we get ImzGν(z) > 0 forz ∈C−. It follows that the
functions in front of the integrals in the definitions ofgandhare
bounded as functions of x, and therefore g and h are
square-integrable. Sincez−^1 yis bounded too, we see that Equation

(1.7.6) defines a bounded operator.
Let us now check that the operator defined in (1.7.6) is the inverse
ofz−

√

SxMy

√
Sx.
Using the notation of the previous subsection, we can writeSx
also asSx=Mx− 1 P 2 + 1 =MxP 2 +P 2 ⊥, whereP 2 ⊥is the projection
onto the orthogonal complement of the subspace of functions that
do not depend on√ y. Its square root can be written as

Sx=M√xP 2 +P 2 ⊥=M√x− 1 P 2 + 1 ; it acts as

(√

Sxψ

)

(x,y) =

(√

x− 1

)∫

R+

ψ(x,y)dν(y) +ψ(x,y)

on a functionψ∈Dom

√

Sx⊆L^2 (R+×R+,μ⊗ν).

Sincehdoes not depend ony, we have

√

Sxh=

√
xh. Forgwe
get
(√
Sx

g

z−y

)

(x) = (

√

x− 1 )

∫

R+

g(x)

z−y

dν(y) +

g(x)

z−y

=

(

(

√

x− 1 )Gν(z) +

1

z−y

)

g(x).

Setφ=ψz−+yg+h. Applying

√

Sxtoφ, we get

(√

Sxφ

)

(x,y)

=

ψ(x,y)

z−y

+

√

x−x

(z−y)

(

( 1 −x)zGν(z) +x

)

∫

R+

ψ(x,y)dν(y)

+

z(x− 1 )

(z−y)

(

( 1 −x)zGν(z) +x

)

∫

R+

ψ(x,y)

z−y

dν(y)

=

ψ(x,y) +g(x)

z−y

.

From this we get
((
z−

√

SxMy

√

Sx

)

φ

)

(x,y) = ψ(x,y)