Hyperreflexivity of Toeplitz analytic operators on the polydisc
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52
pages
English
Documents
Obtenez un accès à la bibliothèque pour le consulter en ligne En savoir plus
- complex hilbert space
- alg lata ?
- vladimir müller
- reflexive df??
- alg lata
- operators
- toeplitz analytic
Publié par
Langue
English
VLADIMIRMÜLLERandMAREKPTAK
Lille,May31June4,2010
HyperreexivityofToeplitzanalyticoperators
onthepolydisc
srotarepocitylanaztilpeoTfoytivixeerrepyHKATPKERAMdnaRELLÜMRIMIDALVKATPKERAMdnaRELLÜMRIMIDALVsrotarepocitylanaztilpeoTfoytivixeerrepyH
srotarepocitylanaztilpeoTfoytivixeerrepyHKATPKERAMdnaRELLÜMRIMIDALVT
=
{
T
ϕ
:
ϕ
∈
L
∞
(
T
)
}A
(
D
n
)=
{
T
ϕ
:
ϕ
∈
H
∞
(
D
n
)
}
T
=
n
{
A
∈
B
(
H
2
(
D
n
)):
S
j
∗
AS
j
=
A
,
j
=
1
,...,
n
}
A
(
D
)=
A
(
S
1
,...,
S
n
)
(
S
i
f
)(
z
)=
z
i
f
(
z
)
for
f
∈
H
2
(
D
n
)
,
i
=
1
,...,
n
T
unitcircle,
D
unitdisc,
D
n
polydisc,
B
n
unitball
H
2
(
D
n
)
⊂
L
2
(
T
n
)
,
H
∞
(
D
n
)
⊂
L
∞
(
T
n
)
P
H
2
(
D
n
)
:
L
2
(
T
n
)
→
H
2
(
D
n
)
ϕ
∈
L
∞
(
T
n
)
T
ϕ
f
=
P
H
2
(
D
n
)
(
ϕ
f
)
for
f
∈
H
2
(
D
n
)
Toeplitzoperatorsonthepolydisc
KATPKERAMdnaRELLÜMRIMIDALVsrotarepocitylanaztilpeoTfoytivixeerrepyH
KATPKERAMdnaRELLÜMRIMIDALVsrotarepocitylanaztilpeoTfoytivixeerrepyHT
=
{
T
ϕ
:
ϕ
∈
L
∞
(
T
)
}A
(
D
n
)=
{
T
ϕ
:
ϕ
∈
H
∞
(
D
n
)
}
T
=
n
{
A
∈
B
(
H
2
(
D
n
)):
S
j
∗
AS
j
=
A
,
j
=
1
,...,
n
}
A
(
D
)=
A
(
S
1
,...,
S
n
)
(
S
i
f
)(
z
)=
z
i
f
(
z
)
for
f
∈
H
2
(
D
n
)
,
i
=
1
,...,
n
T
unitcircle,
D
unitdisc,
D
n
polydisc,
B
n
unitball
H
2
(
D
n
)
⊂
L
2
(
T
n
)
,
H
∞
(
D
n
)
⊂
L
∞
(
T
n
)
P
H
2
(
D
n
)
:
L
2
(
T
n
)
→
H
2
(
D
n
)
ϕ
∈
L
∞
(
T
n
)
T
ϕ
f
=
P
H
2
(
D
n
)
(
ϕ
f
)
for
f
∈
H
2
(
D
n
)
Toeplitzoperatorsonthepolydisc
srotarepocitylanaztilpeoTfoytivixeerrepyHKATPKERAMdnaRELLÜMRIMIDALV
srotarepocitylanaztilpeoTfoytivixeerrepyHKATPKERAMdnaRELLÜMRIMIDALVKATPKERAMdnaRELLÜMRIMIDALVsroT
=
{
T
ϕ
:
ϕ
∈
L
∞
(
T
)
}A
(
D
n
)=
{
T
ϕ
:
ϕ
∈
H
∞
(
D
n
)
}
T
=
n
{
A
∈
B
(
H
2
(
D
n
)):
S
j
∗
AS
j
=
A
,
j
=
1
,...,
n
}
A
(
D
)=
A
(
S
1
,...,
S
n
)
t(
S
i
f
)(
z
)=
z
i
f
(
z
)
for
f
∈
H
2
(
D
n
)
,
i
=
1
,...,
n
aT
unitcircle,
D
unitdisc,
D
n
polydisc,
B
n
unitball
H
2
(
D
n
)
⊂
L
2
(
T
n
)
,
H
∞
(
D
n
)
⊂
L
∞
(
T
n
)
P
H
2
(
D
n
)
:
L
2
(
T
n
)
→
H
2
(
D
n
)
ϕ
∈
L
∞
(
T
n
)
T
ϕ
f
=
P
H
2
(
D
n
)
(
ϕ
f
)
for
f
∈
H
2
(
D
n
)
rToeplitzoperatorsonthepolydisc
epocitylanaztilpeoTfoytivixeerrepyH
arepocitylanaztilpeoTfoytivixeerrepyH(Sarason,Halmos)
fdA
is
reexive
⇐⇒A
=
AlgLat
Lat
A
=
{L⊂H
:
A
L⊂L
forall
A
∈A}
AlgLat
A
=
{
B
∈
L
(
H
):
Lat
A⊂
Lat
B
}
A⊂
AlgLat
A⊂
L
(
H
)
H
complexHilbertspace
B
(
H
)
algebraofallboundedlinearoperators
A⊂
B
(
H
)
subalgebrawith
I
Reexivity
srotarepocitylanaztilpeoTfoytivixeerrepyHKATPKERAMdnaRELLÜMRIMIDALVAKATPKERAMdnaRELLÜMRIMIDALVsrot
srotarepocitylanaztilpeoTfoytivixeerrepyHKATPKERAMdnaRELLÜMRIMIDALVAKATPK(Sarason,Halmos)
fdA
is
reexive
⇐⇒A
=
AlgLat
ELat
A
=
{L⊂H
:
A
L⊂L
forall
A
∈A}
AlgLat
A
=
{
B
∈
L
(
H
):
Lat
A⊂
Lat
B
}
A⊂
AlgLat
A⊂
L
(
H
)
RH
complexHilbertspace
B
(
H
)
algebraofallboundedlinearoperators
A⊂
B
(
H
)
subalgebrawith
I
AReexivity
MdnaRELLÜMRIMIDALVsrotarepocitylanaztilpeoTfoytivixeerrepyH
KATPKERAMdnaRELLÜMRIMIDALVsrotarepocitylanaztilpeoTfoytivixeerrepyH(Sarason,Halmos)
fdA
is
reexive
⇐⇒A
=
AlgLat
Lat
A
=
{L⊂H
:
A
L⊂L
forall
A
∈A}
AlgLat
A
=
{
B
∈
L
(
H
):
Lat
A⊂
Lat
B
}
A⊂
AlgLat
A⊂
L
(
H
)
H
complexHilbertspace
B
(
H
)
algebraofallboundedlinearoperators
A⊂
B
(
H
)
subalgebrawith
I
Reexivity
srotarepocitylanaztilpeoTfoytivixeerrepyHKATPKERAMdnaRELLÜMRIMIDALVA
srotarepocitylanaztilpeoTfoytivixeerrepyHKATPKERAMdnaRELLÜMRIMIDALVThesmallestconstant
k
iscalledthe
hyperreexiveconstant
and
denotedby
κ
A
.
α
(
A
,
A
)=
sup
k
P
⊥
AP
k
:
P
∈
Lat
A
=
=
sup
|h
Ax
,
y
i|
:
k
x
k
=
k
y
k
=
1
,
h
Tx
,
y
i
=
0forall
T
∈A
A
is
hyperreexive
iffthereis
k
suchthatdist
(
A
,
A
)
6
k
α
(
A
,
A
)
Denition(Arveson)
d
α
i
(
s
A
t
,
(
A
A
,
)
A
6
)
d
=
isitn
(
f
A
{
,
k
A
A
)
−
T
k
:
T
∈A}
α
(
A
,
A
)=
sup
d
(
Ax
,
A
x
):
x
∈H
,
k
x
k
=
1
A⊂
B
(
H
)
A
∈
B
(
H
Hyperreexivity
)KATPKERAMdnaRELLÜMRIMIDALVsrotarepocitylanaztilpeoTfoytivixeerrepyH
)KATPKERAMdnaRELLÜMRIMIDALVsrotarepocitylanaztilpeoTfoytivixeerrepyHThesmallestconstant
k
iscalledthe
hyperreexiveconstant
and
denotedby
κ
A
.
α
(
A
,
A
)=
sup
k
P
⊥
AP
k
:
P
∈
Lat
A
=
=
sup
|h
Ax
,
y
i|
:
k
x
k
=
k
y
k
=
1
,
h
Tx
,
y
i
=
0forall
T
∈A
A
is
hyperreexive
iffthereis
k
suchthatdist
(
A
,
A
)
6
k
α
(
A
,
A
)
Denition(Arveson)
d
α
i
(
s
A
t
,
(
A
A
,
)
A
6
)
d
=
isitn
(
f
A
{
,
k
A
A
)
−
T
k
:
T
∈A}
α
(
A
,
A
)=
sup
d<
