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###$ %
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A Remark on Invariant Subspaces of Index One in p
t( )-Spaces
K. Hedayatian*
Department of Mathematics, College of Sciences, Shiraz University, Shiraz 71454, Iran
E-mail: [email protected]
(Received on: 26-06-11; Accepted on: 06-07-11)
__________________________________________________________________________
ABSTRACTI
n this paper, isometric equivalence of the multiplication operator Mz restricted to the invariant subspaces of index 1 in pt (µ) are investigated. Also, the reducing subspaces of the operators Mzi, i 1 on p2 (µ) are discussed.2000 Mathematics Subject Classification: 47B38, 47A15.
Key words and Phrases: Isometric equivalence of subspaces, pt(µ), index 1, reducing subspaces. This research
was in part supported by a grant from Shiraz University Research Council.
_________________________________________________________________________________________
1. INTRODUCTION:
D
1
≤ < ∞
t
p
t( )
µ
( )
t
L
µ
z
∈
( )
t
p
µ
c
( )
| ( ) |
||
||
tL
p z
c
p
µ
≤
p
(
p
t( ))
µ
z
∈
M
r
( )
| ( ) |
||
||
tL
p w
M
p
µ
≤
|
w
−
z
|
<
r
z
( )
p
µ
(
p
t( ))
µ
(
p
t( ))
µ
(
p
t( ))
µ
!
( )
µ
=
D
,
µ λ
({ })
=
0
λ
∈
D
abpe p
(
t( ))
µ
=
D
p
t( )
µ
( )
t
p
µ
"
p
t( )
µ
λ
∈
D
f
∈
p
t( )
µ
λ
∈
D
( )
0
f
λ
=
f z
( ) /(
z
−
λ
)
∈
p
t( )
µ
#z
( )
p
µ
S
p
t( )
µ
( )
t
M
⊆
p
µ
SM
⊆
M
"S
−
λ
I
λ
∈
D
M
p
t( )
µ
(
S
−
λ
I M
)
/(
)
M
S
−
λ
I M
λ
∈
D
$ %M
#&'# (µ
1
A
π
=
)D
( )
t
p
µ
*L
ta---
# $ % ! !&
' ( $ ) ( ) * !+
2. ISOMETRICALLY EQUIVALENT INVARIANT SUBSPACES
(
U
||
Uf
|| ||
=
f
||
dom U
( )
$M
N
p
t( )
µ
+U M
:
→
N
|
M|
NUS
=
S
U
* ,z
M
z -H
2 +-
U
.M
N
+ (/01 *
L
2a ++ #
N
M
zD
N
⊆
M
M
=
N
& '
M
N
+p
t( )
µ
2
N
⊆
M
M
=
N
:
U M
→
N
US
|
M=
S
|
NU
λ
∈
D
K
λMker( |
S
M−
λ
)
∗f
( )
λ
=
f K
,
λMf
∈
M
(
)
{
: ( )
0
Z M
=
λ
∈
D f
λ
=
f
∈
M
}
3 2 0 2 /31\
(
)
D Z M
λ
∈
ker( |
S
M−
λ
)
∗=
C K
.
λM 'ker( |
S
N−
λ
)
∗=
C K
.
λN\
(
)
\
(
)
D Z N
D Z M
λ
∈
⊆
'U
∗ker( |
S
N−
λ
)
∗ker( |
S
M−
λ
)
∗( )
N M
U K
∗ λ=
ϕ λ
K
λλ
∈
D Z N
\
(
)
$(
)( )
,
,
, ( )
( )
,
( ) ( )
N N M
M
Uf
Uf K
f U K
f
K
f K
f
λ λ λ
λ
λ
ϕ λ
ϕ λ
ϕ λ
λ
∗
=
=
=
=
=
f
∈
M
λ
∈
D Z N
\
(
)
'
N
≠
{0}
Z N
(
)
D
T
$D Z N
\
(
)
f
)M
f
)n
z
∈
D
\ (
N
)
ϕ
f
)m
z
(
j
−
1)
n
≤
jm
j
≥
1
1
(
)
(
)
j j j
f
−f
ϕ
=
f
ϕ
f
ϕ
jD Z N
\
(
)
$1
1, 2, 3,
,
j
m
j
j
n
−
≤
=
j
→ ∞
4f
f
ϕ
ϕ
=
D Z N
\
(
)
f
% )M
'||
U
|| 1
=
| (
ϕ
n( )( ) | |
f
λ
=
U f K
n,
λM| ||
≤
f
|| ||
K
λM||
1
n
≥
λ
∈
D Z N
\
(
)
1/ 1/
| ( ) || ( ) |
ϕ λ
f
λ
n≤
(||
f
|| . ||
K
λM||)
n.
n
→ ∞
| ( ) | 1
ϕ λ
≤
λ
f
( )
λ
≠
0
$| ( ) | 1
ϕ λ
≤
2D Z N
\
(
)
%
D
-(
Ug
)( )
λ
=
ϕ λ
( ) ( )
g
λ
g
∈
M
D
λ
∈
5Ω =
{ :| ( ) | 1}
z
ϕ
z
<
$# $ % ! !&
' ( $ ) ( ) * !!
|
ϕ
|
np|
f
|
pd
µ
|
f
|
pd
µ
,
n
1.
Ω
=
Ω≥
|
f
|
pd
µ
0
Ω
=
6 +1
{
:| ( ) |
},
1
n
z
f z
n
n
Ω =
∈ Ω
≥
≥
1
(
)
|
|
|
|
0.
n
p p
n
p
f
d
f
d
n
µ
µ
µ
Ω Ω
Ω
≤
≤
=
'
f
)Ω =
0{
z
∈ Ω
: ( )
f z
=
0}
µ
(
Ω
0)
=
0
-0 0
( )
(
n)
(
n)
0.
n n
µ
µ
µ
∞ ∞
= =
Ω ≤
Ω
≤
Ω
=
$
| ( ) | 1
ϕ
z
=
[ ]
µ
$ψ
( ) | ( ) |
z
=
ϕ
z
D
0(
z
)
1
ψ
≠
z
0D
ψ
( )
z
≠
1
z
0sup
p
µ
=
D
$| ( ) | 1
ϕ
z
=
z
∈
C
ϕ
-
M
⊆
N
! +
' ( '
µ
(
∂
D
)
>
0
#N
) +( )
t
p
µ
N
⊆
M
$M
=
N
* $
A
/21M
N
% 2 $ 2! "
p
t( )
µ
% 2 7% /318 *
L
ap,
p
≥
1
2p a
L
% 2 ({
M
γ γ ∈Γ}
( )
t
p
µ
γγ
µ
∈Γ
∨
M
γ' ( '
M
N
p
t( )
µ
% 2{0}
M
∩
N
≠
$ # 9 +M
∨
N
* 6 0 23 /31
M
∨
N
% 2 ' $ 2' ( )
{
M
γ γ ∈Γ}
p
t( )
µ
% 20
M
γ)
0
M
γ∨
M
γ % 2γ
∈ Γ
γ
1∈ Γ
1
M
γM
γγ ∈Γ
∨
+
1
M
γM
γγ ∈Γ
=
∨
* $ 0 20 7 8 /31
M
γ# $ % ! !&
' ( $ ) ( ) * !,
' ( '
{
M
γ γ ∈Γ}
p
t( )
µ
% 20
M
γM
γγ ∈Γ
+
0
M
γM
γγ ∈Γ
=
* $ 0 2: /31
M
γγ ∈Γ
% 2 9 $ 2
N
p
t( )
µ
N
⊥N
(
p
t( ))
µ
∗{
(
t( )) : ( )
0,
}
N
⊥=
x
∈
p
µ
∗x g
=
∀ ∈
g
N
; +' ( * '
S
σ
( )
S
=
D
M
N
+
p
t( )
µ
N
⊆
M
N
% 2 #%
λ
(
|
)
N
S
σ
∗
⊥ $M
=
N
6 < = /31
M
% 2 $ 23. REDUCING SUBSPACES
1
i
≥
M
'p
2( )
µ
' 'M
⊥⊆
M
⊥S
'( )
µ
⊆
D
z
n,
z
m=
0
n
≠
m
.,.
2
( )
p
µ
%v
/> 21µ
D
1
(
)
( ),
2
i
d
µ
re
θd dv r
θ
π
=
,
0,
n m
z
z
=
n
≠
m
#||1||
L2( )µ=
1
$ %,
1
i
S
i
≥
&
2
1/ 2 ( )
||
n||
,
1
n L
w
z
n
µ
=
≥
1
lim
nsup
n1.
n n
w
w
w
+=
=
$
S
z
p
2( )
µ
#i
>
1
?
7 8
m n
,
0
≤
n
≤ −
i
1
0
≤
m
≤ −
i
1
%
k
>
0
n ki m ki
n m
w
w
w
w
+
≠
+i
S
%2
i# $ % ! !&
' ( $ ) ( ) * !&
n ki m ki
n m
w
w
w
w
+
=
+" @ >
S
i$
2
,
0
n
n n
z
e
n
w
=
≥
p
2( )
µ
'2
1 1
(
/
)
n n n n
Se
=
w
+w
e
+S
+V
n=
(
W
n+1/
W
n)
2 $6 3 /<1
S
!w
0=
1
*5 A /<1
S
+M
z2
0 1 2
, (
{
,
,
, })
w
H
w
=
w
w w
0
( )
nn
f n z
∞=
2
0
| ( ) |
nn
f n
w
∞
=
< ∞
$0
||
|| sup
n1,
n
S
v
≥
=
=
2 2
0
1
n
w
≤
w
=
n
≥
0
$2 2
1
=
lim
nw
n≤
lim
nw
n≤
1
n n
w
2n
→ +∞
9 20
( )
( )
n wn
f z
f
n z
H
∞
=
=
∈
2 2
lim | ( ) |
nf n
=
lim | ( ) |
nf n
w
n≤
1.
6 +
lim | ( ) | 1
nf n
≤
-f z
( )
D
9 $B
C
/B1+ ,
(
)
1
( )
2
i
d
µ
re
θd dm r
θ
π
=
m
/> 217 8 $ 3 -
i
>
1
S
i %2
iREFERENCES
[1] Aleman A., Richter S., Sundberg C., Nontangential limits in pt(µ)- spaces and the index of invariant subspaces, Annals of Mathematics, 169 (2009), 449-490.
[2] Richter S., Invariant subspaces in Banach spaces of analytic functions, Trans, Amer. Math. Soc. 304 (1987), 585-615.
[3] Richter S., Unitary equivalence of invariant subspaces of Bergman and Dirichlet spaces, Pacific J. Math, 133 (1988), 151-156.
[4] Shields A. L., Weighted shift operators and analytic function theory, Math. Surveys, Vol. 13, Amer. Math. Soc., Providence, 1974.
[5] Stessin M., Zhu K., Reducing subspaces of weighted shift operators, Proc. Amer. Math. Soc. 130 (2002), 2631-2639.
