Publicacions
Ma emá iques,
,
Vol
34
(1990),
181-197
.
Abs ac
FOURIER
ANALYSIS
OF
A
SPACE
OF
HILBERT-SCHMIDT
OPERATORS-
NEW
HA-PLITZ
TYPE
OPERATORS
JAAK
PEETRE
I
a
g oup
ac s
ia
uni a y
ope a o s
on
a
Hilbe
space
o
unc ions
hen
his
g oup
ac ion
ex ends
in
an
ob ious
way
o
he
space
o Hilbe -
Schmid
ope a o s
o e
he
gi en
Hilbe
space
.
E en
i
he
ac ion
on
unc ions
is
i educible,
he
ac ion
on
H
.S
.
ope a o s
need
no
be
i e-
ducible
.
I is
o en
o
conside able
in e es
o
ind
ou
wha
he
i e-
ducible
cons i uen s
a e
.
Such
an
a i ude
has
ecen ly
been
ad oca ed
in
he
heo y
o
"Ha-pli z"
(Hankel
+
Toepli z)
ope a o s
.
In
his
pape
we
sol e
his
p oblem
he
space
o
H
.S
.
ope a o s
o e
he
Hilbe
space
L
Z
(A,
p
.) o
squa e
in eg able
unc ions
o e
he
uni disk
A
equipped
wi h
he
Dzh bashyan
measu e
dpa
(z)
=
(ce+1)(1-jzj')°dA(z)(a
>
-1)
.
This
complemen s
he
ea lie
esul s
.
In pa icula
we
disco e
many
new
Ha-pli z
ype
ope a o s
.
The
ques ion
o hei
smoo hness
p ope ies
(S
p-
es ima es
e c
.)
is
howe e
only
ouched
upon
.
In oduc ion
I
a
g oup
ac s
ia
uni a y ope a o s
on
a
Hilbe
space
o
unc ions
hen
his
g oup
ac ion
ex ends
in
an
ob ious
way
o
he
space
o Hilbe -Schmid
(H
.S
.)
ope a o s
o e
he
gi en
Hilbe
space
.
(Mo e
gene ally,
one can
conside
H
.S
.
ope a o s
om
one
Hilbe
space
in o
ano he
wi h
a
di e en
g oup
ac ion
on
each
o
hese
spaces
.)
E en
i
he ac ion
on
unc ions
is
i educible,
he
ac ion
on
H
.S
.
ope a o s
need
no
be
i educible
.
I is
o en o
conside able
in e es
o ind
ou
wha
he
i educible
cons i uen s
a e
.
Such an
a i ude
has
ecen ly
been
ad oca ed
in
he
heo y
o
"Ha-pli z"
(Hankel
-}-
Toepli z)
ope a o s
.
Example
1
.
Conside
he spaces
A
=
A''2(A)
(de ini ion
in
Sec
.
1),
A
1
=
he
o hogonal
complemen
o
A
in
L
2
(A,pa)
(de ini ion
in
Sec
.
1),
A =
he
space
consis ing
o
all
conjuga es
o
he unc ions
in
A
.
Ope a o s
om
A
in o
A
ha e
been
s udied
om
his
poin
o
iew
in [JP1]
("small"
Hankel
ope a o s
o
highe
weigh )
.
Simila ly,
ope a o s
om
A
in o
A
1
ha e been
s udied
in
[BJP]
("big"
di os)
.
In
bo h
cases
he
g oup
is
he
p ojec i e
g oup
o
con o mal
sel maps
o
he
uni
disk
A
in
C
o
a
double
co e
.
18
2

J
.
PEETRE
Example
2
.
The
s udy
o
he
case
when
A
is
eplaced
by
he
uni
ball
in
Cn
has
been
ini ia ed in
[P1]
.
Example
3
.
H
.S
.
ope a o s
on
L
2
(Rn)
unde
he ac ion
o
he
"ax+b"-g oup
(dila ions
+
ansla ions)
a e
analyzed
in
[JP2],
[P2]
(so-called
"pa acommu-
a o s")
.
In
his
pape
we wan
o
decompose
he
space
o
H
.S
.
ope a o s
o e
L
2
(A,
m«)
.
This
complemen s
he
esul s
o
[JP1], [BJP]
.
In pa icula ,
we
disco e
many
iew
Ha-pli z
ype
ope a o s
.
The
ques ion
o he
smoo hness
p ope ies
(S
p
-
es ima es
e c
.)
is
howe e
only
ouched
upon
.
The e
a e
good
hopes
ha
one
could
be
able o
play
he
same
game
wi h
o he
symme ic
domains
(in
highe dimensions),
in
he
i s
place
he
ball
(c
.
The
o ganiza ion
o
he
pape
is
as
ollows
.
Sec
.
1
in oduces
he
no a ioíi
.
In
Sec
.
2
we
conside a ious
in a ian
measu es
on
he
space
o
H
.S
.
ope -
a o s
o e
L
2
(A,íc
a
)
.
In
pa icula ,
his
allows
o
w i e
he
space
in
ques ion
as a
enso
p oduc
whe e
one
ac o
is
he
space
L
2
(G),
G
being
he
g oup
o
all
con o mal
se1 maps
o
A
.
The
esul s
o
his
Sec ion
a e
o
independen
in e es
and
we
in end o
e um
o
hem
on
a
subsequen
occasion
.
In
Sec
.
3
we
hen
apply
Planche el's
heo em
in
he
la e
space
o
p oduce
he
desi ed
decomposi ion
o
L2
(A
,
ca)
.
Sec
.
4
is
de o ed
o
some
examples
showing
how
known
ype
o (big)
Hankel
ope a o s
[JP1],
[BJP]
(Example
1
ul a)
i
in o
ou
new
scheme
.
In
Sec
.
5,
e u ning
o
he
gene al
case,
we
w i e
down
he
exp ession
o
he
H
.S
.-no m
o
ou
new
Hankel
ope a o s
.
Sec
.
6
b ie ly
ouches
upon
he
S
p
- heo y,
s ill
in
i s
emb yo
.
In
Sec
.
7
we
ske ch
an
al e na i e
in ini esimal
app oach
o
ou
p oblem
.
I
is
no
en i ely
success ul,
as
we
ha e
no
been
able
o
ca y
ou he
spec al
analysis
o
he
Casimi
ope a o
encoun e ed
.
In
Sec
.
8
we
specialize his
o
he
case
o
ope a o s
om
A"'2(A)
in o
i sel
(gene alized Toepli z
ope a o s)
.
Finally,
he e
is
an
appendix
o
which
we
ha e
de e ed
some
(dull?)
calcula-
ions
pe aining
o
Sec
.
2
.
We
in end
o
apply
his
esul
in
a
di e en
con ex
in
a
subsequen
publica ion
.
Acknowledgmen
.
I
am
g ea ul o
he
e e ee
o his
many
commen s,
he
schola ly
ones
and
also
he
humo is ic
ones
.
He
also
de ec ed
a
missing
ac o
1/(167
2
)
in
o mula
(6),
Sec
.
2
.
In
addi ion,
I
am
obliged
o
Richa d
Askey
and
And é
Un e be ge
o
p ecious
in o ma ion
.
1
.
Mainly
no a ion
Le
us hus
begin
by
ixing
some
no a ion
:
FOURIER
ANALYSIS
OF
HILBERT-SCHMIDT
OPERATORS

18
3
0
-
uni disk
in
he
complex
plane
C
.
T
-
i s
bounda y
( he uni
ci cum e ence)
.
dxdy
idzdz
dA(z)
=

7

=

2-

-
no malized
Euclidean
a ea
measu e on
0
.
dI(z)
_
(1dA(I))2
-
in a ian
(Poinca é)
measu e
on
A
.
dp
a
(z)
=
(a
+
1)(1
-
Iz1
2
)°dA(z)
(a
>
-1)
-
Dzh bashyan measu e
.
L
2
(
0
,
[ ,,)
=
he
space
o
squa e
in eg able
unc ions
wi h
espec
o
di
a
.
Aa~2(A)
=
Hol(A)
n
L
2
(á,
/la)
-
Dzh bashyan
(o
weigh ed
Be gman)
space,
he
space
o
holomo phic
unc ions
in
.L2(0,
pa)
.'
The
co esponding
no m
and
he
co esponding
inne
p oduc
will
be
w i en
11
-
IIa
and
(
.,
.)a
espec i ely
.
The
space
o
H
.
S
.
ope a o s
on
L
2
(A,
p,~)
can
be
iden i ied
wi h
L
2
(0,
pa
)
L
2
(A,
/
a
)
=L
2
(A
x
A,
ha),
ha
is,
a
H
.S
.
ope a o
T
=
TF
on
L
2
(
0
,
/~
a)
iS
gi en
by
a
ke nel
F(z
l ,
z
2
)
such
ha
IITFII
2
=
LA
IF(zI,z2)I
2
dha(zl)dUa(z2)
<
W-
G=
SU(1,1)
-
MSbius
g oup
.
G
=
PSU(1,1)
=
SU(1,1)/
i
1 -
p ojec i e
g oup
;
he
elemen s
x
o
G
a e
hus
ac ional
linea
unc ions
o
he
ype
x(z)
=
e
z
+
d
,
whe e
(
c

d
)
is
in
G, uniquely
de e mined
by
x
up
o
sign
.
G
-
uni e sal
co e ing
g oup
o
G
o
G
.
I
will
be
necessa y
o
dis inguish
ca e ully
be ween
hese
g oups
.
An
elemen
o
G
is
an
elemen
x
o
G
plus
a
choice
o
a
de e mina ion
o
he
squa e
oo
P
.
Simila ly,
an
elemen
o
G
is
an
elemen
o
G
plus
a
choice
o
a
de e mina ion
o
he
loga i hm
log
x'
( hen
we
can
de ine
a bi a y
powe s
o x')
.
dH(x)
deno es
he
Haa
measu e
on any
o
hese
g oups
.
The
g oup
G
ac s
on
elemen s
(z) o
L
2
(O,i¿a)
o
A"
,2
(A)
acco ding
o
he
ule
(z)
H
(x(z))(x
(z))

(=
(
áz
+
d
)(cz
+
d)-(«+2)1
Tha
hese
g oup
ac ions
a e
uni a y
ollows
om
he
o mula
dí a(x(z))
=
IXI(z)I«+2dMa(z)
.
(x
EG)
.
'In
[JPR],
p
.
63
his
space
was
called
he
Ba gmann-Be gman-Beso -Dzh bashyan-Fishe -
Fock-Segal
space
.
As
he
e e ee sugges s,
one
could
as well
ha e
included
he
na es
Pe e sson
and
Hecke
oo
.
The
eason
why
we
ha e
gi en
p e e ence
o
he
na e
Dzh bashyan
he e
is
ha
we
hink
ha hi he o
one
has
no
in
he
li e a u e
payed
su icien
ibu e o
he
achie emen s
o
he
A menian
school
o
analysis
.
18
4

J
.
PEETRE
Simila ly,
a
ke nel
F(zl,
z2)
expe iences
he
change
F(zi,zz)
i
--
>
F(x(zl),x(z2»(x
' (
zl»
241
(
77
(z2»
241

(x
E
G)
.
Ou
main
conce n
is
hus
o
decompose
he
space
S2(L
2
(0,
p,
«
))
o
H
.S
.
ope -
a o s
unde
he
las
ac ion
.
Begin
by
w i ing
2
F(zl,
z2)
=
K«(zl,
zz)Fo(zi
)
z2)
whe e
K«(zl,
z2)
_
(1-zlz2)-(«+2)
i
s
he
ep odcing
ke nel
in
A«,
z(
A)
.
Then
Fo
ans o ms
as
Fo(ZJ,z2)
-
Fo(x(zi),x(z2))

(x
E
GSlc!),
his in
iew
o
he
well-known
o mula
(
2
)

K«(x(zl),x(z2)»'(zj)

(!
7
(
-
z
-
2»
42
=K«(zi~zz)

(~
E
G)
.
+
2

2
We
ag ee
o
e e
o
Fo
as
he
" educed"
ke nel
.
Thus
we
may
wo k
wi h
Fo
ins ead
o
F
.
In
e ms
o
Fo
he
H
.S
.
no m
is
gi en
by
IITFII
2
=

(1
-
I
zi
l
2 )(
1
-
2Iz21 2
)
/
«+2
IFO(zi
z2)I
2
dI(z1)dI(z2)-
A
(

I1-ziz2l
I s
in a ian
cha ac e
is
hus appa en
.
Clea ly
2
.
In a ian
measu es
dJ=dI0dI
is
an
in a ian
measu e on
A
x0
.
Bu ,
as
he
g oup
G
does
no
ac ansi i ely
on
his
se ,
i
is
no
unique
.
The e
a e
o he
in e es ing
in a ian
measu es
on
A
x
0
o
a he
on
A
x
A
(diagonal),
he
mani old
o
all
o de ed
pai s
o
elemen s
o
A
(o
o ien ed
hype bolic
segmen s)
.
2
0ne
mo i a ion
o
his
cons uc ion
comes
om
Be ezin's
p og am
o
quan iza ion
(c
.
[P3])
.
The
e e ee
commen s
ha
" he
idea o
ac o iza ion
o
au omo phic
o ms
goes
back
o
many,
many
people,
p obably
much
be o e
Be ezin
(Poinca é,
o
example)"
.
Le
hus
an
o de ed
pai
(z
1 ,
z
2 ),
wi h
z
1
E
0,
z
2
E
0, be
gi en
.
Conside
he
unique
(o ien ed)
geodesic
h ough
z1
and
z2,
di ec ed
om
z2
o
z1
.
The e
is
a
unique
ans o ma ion
1
in
G
such
ha
i s
p eimage
is
he
in e al
(-1,1)
and
such
ha
he
p eimages
o
z
1
and z
2
a e
wo
symme ically
si ua ed
(abou
he
o igin 0)
poin s
p1
and
P2
on
(0,1)
and
(-1,
0)
espec i ely
:
(4)

z1
=
«PI),
z2
=
E(p2)

(1
%
pl
=-
p2
%
0)
.
Le
A
deno e
he
exponen ial
o
he
hype bolic
dis ance
be ween
z
1
and
z
2
(Le
.
log A
_

Ps

dx

_
1
log
1
+p1
-
1
log
1
_

+p2
=
log
1
+pl
o

1
+pl
~

-

a
=

)
.
n1
1-
x
2
2
1-
pl 2
1
-
p2

1
-
p,

1
-
pl
Thus
he
pai
(z1,
z2)
is
uniquely
de e mined
by
he
pai
(1,
A),
and
he
mani old
o
all
o de ed
pai s
ge s
iden i ied
wi h
he
p oduc
G
x
R+
.
No ice
ha
G
ac s
on
he
i s
ac o
ia
mul iplica ion
om
he
le
(see (4))
and
i ially
on
he
second
ac o
( he
hype bolic
dis ance
is
p ese ed),
Le
.
o
(x(z1),x(z2)),
whe e x
E
G,
he e
co esponds
he pai
(x
j,
A)
.
I
ollows
ha
an
in a ian
measu e
on
he
se
0
x
0
(diagonal)
is
gi en
by
FOURIER
ANALYSIS
OF
HILBERT-SCHMIDT
OPERATORS

18
5
We
can
u he
iden i y
he
g oup
G
i sel
wi h
he
mani old
o
geodesics
equipped
wi h
a
base
poin
.
Le
11
and
S2
be
he
endpoin s
o
he
geodesic
on

18
6

J
.
PEETRE
T
and
le
m
be
he
exponen ial
o
he
hype bolic
dis ance
om
he
base
poin
o
he
Euclidean
midpoin
z* o
he
geodesic
(=
a e o a
Euclidean
o hogonal
ci cle)
.
We
may
ega d
(S1,
(2,
m)
as
coo dina es
o
he
co esponding
g oup
elemen
1
.
Then
he
Haa
measu e
on
G
can
be
w i en
Le
u he
m
i
and
m2
measu e
he
dis ance
om z
1
and
z2
can
be
ew i en
as
_
dJ
_

1

m
2
-
m2
dJ'
16, 2
m1In2
.
Id(,
¡
.
jd(2
j
dm
-
1(
;-
'
(212

m
.
dJ'
-
id(11
-
ld(21

d n
l
dm
2
1S1
-
S2
12

m1

m2
A
compu a ion,
which
I
de e
o an
appendix,
e eals
ha
he
Radon-Nikodym
de i a i e
ó
he
measu e
(3)
wi h
espec
o
he
one
in
(5)
o
(5'),
is
gi en
by
(A
p io i
one can
say
ha
i
mus
be
a
homogeneous
unc ion
o
deg ee
0 in
m1
and
m
2
.)
I
we
pu
A
=
e°,
whe e
he
quan i y
u
hus
has he
in e p e a ion
o
hype bolic
dis ance,
hen
we
ha e
in
(6)
essen ially
he
hype bolic
sine,
sinh
Q,
which
pe haps
looks
mo e
con incing
.
3
.
Planche el
heo em
Fo
he
g oup
G
his
heo em
ells
us
ha
unde
le
mul iplica ion
by
g oup
elemen s
(le
egula
ep esen a ion)
he
Hilbe
space
L
2
(
G) decomposes
in o
an
o hogonal
sum
00 00
=
82
(ñ
-
iX-1)
.1
.
L
2
(G)
=
1
:
®
D

,
®
j
:
®
D

,
®
C
®
C
m-1 m-1
o
z*
.
Then
(5)
( he
case
o
G
is
in
[L]
om which
he
p esen
case
G
easily
ollows)
.
I
consis s
hus
o
a
"disc e e"
pa
and a
"con inuous"
pa
.
He e
I
will
ocus
on
he
disc e e
pa
only,
aking
each
summand
sepa a ely
.
As
D

,
and
D

,
a e
each
o he s
conjuga es
i
su ices
o
consid
e
D

,
.
I is
known
ha
D

,
is
isome ically
isomo phic
e
Aa,2(0)
®
AQ
,2
(0)
whe e
Q
=
2m
-
2
.
No ice
ha
(in
con as
o
G)
o
he
g oup
G
only
e en
deg ees
occu !
Thus
D

,
is
no
i educible
bu
con ains
A,3,2
(A)
wi h
in ini e
mul iplici y
.
In
he
ollowing discussion
i
is
use ul
o
bea
in
ones
mind
he
ollowing
pu ely
algeb aic
ac
-
essen ially
a
consequence
o
Schu 's
lemma
(c
.
[JPW])
.
Lemma
.
Le¡
V
be
any
simple
module
(o en
any
C-algeb a)
and
conside
an
i educible
submodule
W
o
he
di ec
sum
V®V®V
®
.
. .
(wi h
elemns
x1
®
x2
®
.
.
.
)
.
Then
¡he
elemen s
o
W
a e
o
he
o m
plx
®p2x
®
.
.
.
whe e
xE
V
uns h ough
his
module
and
pl,p2,
. . ,
a e
ixed
complex
numbe s
.
The
gene al
o m
o
an
elemen
o
D

,
is
whe e
'¿
hus
is
analy ic
in
he
i s
a gumen and
an i-analy ic
in
he
second
a gumen
.
In
pa icula ,
D

,
con ains
any
unc ion
o
he
o m
wi h
0,0E
A
0,2
(A)
(c
.
[L],
p
.
181)
.
I
we
ix
i
:~
0 and
le
0
a y
we
ge
a
submodule
o
D

,
isomo phic
o
AQ
,2
(A)
.
Choosing
an
o hono mal
basis
{Oj}
in
Aa'2(A)
(e
.g
.
he
s anda d
basis
ob ained
by
no malizing
he
monomials
{zl})
we
decompose
D

,
in o
a
di ec
sum
o
i educible
submodules
.
Nex
pick
an
o hono mal
basis
{Bi}
also
in
he
space
L
2
((1,
oo),
A,,),
whe e
(No ice
ha
hey
a e o
he
o m
FOURIER
ANALYSIS
OF
HILBERT-SCHMIDT
OPERATORS

187
( )
-
J

~( ( ),
~( ~( ))md~a( ),
0
. (1)
=
0(1( ) )(j'( ))mdM#( )
0
2

=

2ñ

)
2(a+2)

dñ
1s~

dA~(a)

52
+
1

(a
-
IX
-1
)
1
.
wi h Oj( )
E
AP
2(A)
.
We
hen
ob ain
( 1
-
Iz1I 2
)(1-Iz2I
2
)
_

2
,

/
2
)
11
-
z1Z21
2
~
-F
1
Then
o
e e y
H
.S
.
ope a o
TF
he
co esponding
educed
ke nel
F
o
(see
Sec
.
1)
admi s
he
decomposi ion
FO(z1,
z2)
_

A=(1)Bi(A)
.
i
Le
us
es ic
ou
a en ion
o
he
case
when
all
he
Ai
belong
o
D

ha
is,
Aj(j( ))
=
~Y'ij(S( ))Y'j( )(S~( ))mdhQ( )
7
FO(z1,z2)
_

~i
;(~( ))
;( )(
~( ))
dl~Q( )Bi(~)
.
i>j
18
8

J
.
PEETRE
In oking
he
lemma
(o
a he he
gene al
philosophy
behind
i )
we
see
ha
he
mos
gene al
i educible
submodule
is
go en
by
aking
all
he
O=i
p opo ional
o
one
and
he
same
unc ion
0
in
A
Q,2
(0),
ha
is,
O
;j
=
pijo
o sui able
scala pi7
.
Such an
i educible
submodule
is
hus
spanned
by
educed
ke nels
o
he
o m
wi h
a( ,
A)
=
Epii0
.i( )Bi(A)
.
i,J
By
change
o
a iable
(see
(1)) his
o mula
can be
w i en
ha
is,
whe e
wi h
(8)
simila ly
o
A
o
:
FO(zl,
z2)
=
L
O(e( ))a( ,
A)(1'( ))'dpp( )
0
Fo(zl,z2)=

( )a(
-1
( ),~)((
_1)~( ))md ~a( ),
0
(7)

Ao(z1,
z2,
)
=
a(17
1
( ),
~)((~-1)~( ))~`
.
Rein oducing
he
ep oducing
ke nel
we
can
inally
w i e
he
ke nel
i sel
as
A(z1,
z2,
)
=
K«(z1,
z2)Ao(z1,
z2,
)
.
No ice
ha
he
ke nel
A
obeys
he
ans o ma ion
law
(c
.
(2))
A(x(zl
),
x(z2),
x( ))
=
A(z1,
z2,
)(x'(z1))241(x7(z2))~
(7'( ))
-m
l
Di ec
check
ha Such
a
ke nel
de ines
an
in a ian
submodule
:
FO(X(zl),
x(z2))
=
O( )Ao(x(zl),
x(z2),
)dpp( )
0
Fo(z1,
21
2)
=

O( )Ao(zl,
z2,
)dpa( )
0
F(z1,
z2)
=
L
O( )A(zl,
z2,
)dPP( )
0
Ao(x(zl),
x(z2),
x( ))
=
Ao(zl,
z2,
)(x~( ))-~
O(x( »Ao(x(z1),
x(z2),
x( »di p(x( »
O(x( ))(x'( ))'Ao(z1,
z2,
)I
X
I
( )I-2'dhp(x( ))
=
o( ))w( ))'Ao(z1,
x2,
)dN p( ),
0
whe e
we
in
he
las
s ep
once
again
in ol ed
(1)
.
Rema k
.
In
hindsigh
we
could
pe haps
ha e
w i en
down
his
o mula
igh a
he beginning,
bu
we
ind
i
mo e
ins uc i e
o
ollow
he
pa h
o
disco e y
.
FOURIER
ANALYSIS
OF
HILBERT-SCHMIDT
OPERATORS

189
4
.
Examples
Example
1
.
m
=
1
.
Con en ional
big
Hankel
ope a o s
[A],
[AFP]
co e-
spond
o
he
ke nel
jz2
F(z1,z2)
=
K
.(z1,z2)
J

~(C)dC
_
i
(I
we
in oduce
he
p imi i e
b
o
0
(Le
.
b'
=
0)
we
ge
he
usual exp ession
F(z1,
z2)
=
K
.(zl, z2)(b(z1)
-
b(z2»-)
Check
o
he co a iance
:
(=1)
Fo(x(zl),x(z2»
=
z(z=2)
=
O( )d
o m
(7)
.
Indeed,
one
has
o ake
z1
-
z2
A
O(
z
1,
z2)
_
(1
-
z1~2(1
-
z2í)2
.
«x( ))x'( )d
.
Bu
his
is
no
o he
o m
(7)
.
I
is
howe e
easy
o
ew i e
he
o mula
in
he
A(z
1
,
z2,
)

(1
-
(12

~~
1
.
z,

1

1
2

Zl
1

1

1

z
1
-
z2
=
C1-z1 -
1
-z2 ~
_
(1-zl~(1-z2
)
.
Example
2
.
m=
2
.
We
ha e
he
(big)
"Lag ange-Hankel"
ope a o
[BJP],
[P1]
co esponding
o
Fo(z1,
z2)
=
b
'
(
zl)
-}-
b'(z2)
-
2
b(z
1
)
-
b(z2)
.
zl
-
z2
To
ge
he
ansi ion
o
ou
p esen
o m
o
no a ion
we
in oke
he
hi d
o de
de i a i e
b"'
=
0
o
he
gi en
b
.
(This
is
in
acco dance
wi h
he
equi emen
o
"Ból's
lemma"
[GP]
.)
Then
we
ind
F+
O
(
zl z2
)
_
j

(z1
-
O(
z
2
-
~)
~(C)dC
x(=2)

zl
-
.
z2
As
in
Example
1
one can
w i e
down
he
co esponding
ke nel
A
o
,
The me hod
in
[P1]
is
howe e
p e e able,
and
applicable
also o
m
>
2
.
19
6

J
.
PEETRE
On
he
o he
hand
Bu
(as
k
-
k(
2
)
(1
Thus
and
2

(
2
ikm1
2
_
1
+
Ik1
2
m
2
-
2Reikm(1(2
IxI

I1
-
ikm1
2
1
+
k2
m
2
-
2Reikm
1
-
1x12
=
2Re(ikm((1(2
-
1))
ik
2
ik(L
-
1)

ik((2
-
(i)(
1
+
S2
1
)
-
2ik((2
-
(1)
.
(1

(2

(1

(1
(2

(1
I
ollows
ha
(1
-
IZ12)2
=
41(1
-
(212m2
(1
-
1x112)2(1
-
1x212)2

I1
16
1(
km11(214mik n2l4
.
i

2

dx
l
dz
l
dx
2
dz2

_

1

m
2
-
_

_

Id(,
1 -
1d(21
_dml
dm2
(27 )

x
(1
-
1x
11
2
)2(
1
-
1x212)2

167
2
m
1
m2

1(1
-
(21
2

m
1
m2
as
p e iously
claimed
(Sec
.
2,
o mula
(6))
.
Re e ences
[A]
S
.
AXLER,
The
Be gman
space,
he
Bloch
space,
and
commu a o s
o
mul iplica ion ope a o s,
DukeMa h
.
J
.
53
(1986),
315-332
.
[AFP]
J
.
ARAZY,
S
.
FISHER
AND
J
.
PEETRE,
Hankelope a o s
on
weigh ed
Be gman
spaces,
Ame
.
Ma h
.
J
.
110
(1988),
989-1053
.
[BJP]
J
.
BOMAN,
S
.
JANSON
AND
J
.
PEETRE,
Big
Hankel
ope a o s o
highe
weigh ,
Rend
.
Ci c
.
Mai
.
Pale mo
37
(1988),
65-78
.
[GP]
B
.
GUSTAFSSON
AND
J
.
PEETRE,
No es
on
p ojec i e s uc u es
on
complex
mani o1ds,
Nagoya
Ma h
.
J
.
116
(1989),
63-88
.
[JP1]
S
.
JANSON
AND
J
.
PEETRE,
A
new
gene aliza ion
o
Hankel
ope a o s
( he
case
o
highe
weigh s),
Ma h
.
Nach
.
132
(1987),
313-328
.
[JP2]
S
.
JANSON
AND
J
.
PEETRE,
Pa acommu a o s
-
boundedness
and
Scha en- on
Neumann
p ope ies,
T ans
.
Am
.
Ma h
.
Soc
.
30
5
(1988),
467-504
.

FOURIER
ANALYSIS
OF
HILBERT-SCHMIDT
OPERATORS

197
[JPR]
S
.
JANSON,
J
.
PEETRE
AND
R
.
ROCHERG,
Hankel
o ms
and
he Fock
space,
Re is a
Ma
.
Ibe o
.Ame
.
3
(1987),
61-138
.
[JPW]
S
.
JANSON,
J
.
PEETRE
AND
R
.
WALLSTÉN,
A
new
look
on
Hankel
o ms
o e
Fock
space,
S udia
Ma h
.
95
(1989),
33-41
.
[L]
S
.
LANG,
"SL
2
(R),"
Addison-Wesley
Publishing
Co
.,
Reading,
Mass
.
-
London
-
Ams e dam,
1975
.
[P1]
J
.
PEETRE,
"Hankel
ke nels o
highe
weigh
o
he
ball,"
Technical
epo ,
Lund,
1988
.
[P2]
J
.
PEETRE,
Pa acommu a o s
-
a
b ie
in oduc ion,
open
p oblems,
Re is a
Ma
.
Uni
.
Complu ense
Mad id
2,
núme o
suplemen a io
( o
ap-
pea )
.
[P3]
J
.
PEETRE,
The
Be ezin
ans o m
and
Ha-pli z
ope a o s,
J
.
Ope a o
Theo y
( o
appea )
.
[P4]
J
.
PEETRE,
Some
unsol ed p oblems, In
:
Colloquie
Ma hema ica
So-
cie a is
Janos
Bolyai,
Al ed
Haa Memo ial
Con e ence,
Budapes
(Hun-
ga y),
711-735,
1985
.
No h-Holland,
Ams e dam,
1986
.
[R]
G
.
RIDEAU,
Su
la
éduc ion
du
p odui
enso iel
des
ép esen a ions
de
la
sé ie disc é e
de
SL(2,
R),
Ann
.
Ins
.
H
.
Poinca é
Seci
.
A
.
4
(1966),
67-76
.
[UU]
A
.
UNTERBERGERAND
J
.
UNTERBERGER,
La
sé ie
disc é e
de
SL(2,
R)
e
les
opé a eu s
pseudo-di é en iels
su
une
semi-d oi e,
Ann
.
Se¡
.
Ecole
No m
.
Sup
.
17
(1984),
83-116
.
[W]
J
.
WILSON,
Some
hype geome ic
o hogonal
polynomials,
SIAM
J
.
Ma h
.
Anal
.
11
(1980),
690-701
.
Keywo ds
.
Hankel
ope a o ,
ep oducing
ke nel,
decomposi ion
o
g oup
ep esen a ions,
Hilbe -Schmid
ope a o
1980
Ma hema ics
subjec
classi ica ions
:
47B35, 30C40, 43A65,
43A80
Ma ema iska
Ins i u ionen
S ockholms
uni e si e
Box
6701
S-113
85
S ockholm
SWEDEN
Rebu
el
16 de
Juny de
1989