Hyperreflexivity of Toeplitz analytic operators on the polydisc

Vladimir Müller And Marek Ptak

Description

Hyperreflexivity of Toeplitz analytic operators VLADIMIR MÜLLER and MAREK PTAK Hyperreflexivity of Toeplitz analytic operators on the polydisc VLADIMIR MÜLLER and MAREK PTAK Lille, May 31 – June 4, 2010 VLADIMIR MÜLLER and MAREK PTAK Hyperreflexivity of Toeplitz analytic operators

• complex hilbert space


• alg lata ?


• vladimir müller


• reflexive df??


• alg lata


• operators


• toeplitz analytic

Sujets

• Hilbert space
• Operator (disambiguation)

Informations

• Publié par: profil-urra-2012
• Nombre de lectures: 40
• Langue: English

Extrait

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<