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AdvancesinAppliedMathematicsA^êÆ?Ð,2021,10(11),3699-3711
PublishedOnlineNovemb er2021inHans.http://www.hanspub.org/journal/aam
https://doi.org/10.12677/aam.2021.1011393
Dirichlet˜mþH-ToeplitzŽf
†5
oooŠŠŠYYY
úô“‰ŒÆêƆOŽÅ‰ÆÆ§úô7u
ÂvFϵ2021c108F¶¹^Fϵ2021c1029F¶uÙFϵ2021c119F
Á‡
©ÄuBergman˜mþH-ToeplitzŽfïħ3ùŸ©Ù¥Ì‡ïÄDirichlet˜mþH-
ToeplitzŽf†5"
'…c
H-ToeplitzŽf§Dirichlet˜m§†5
CommutantsofH-ToeplitzOperators
onDirichletSpace
MengkeLi
InstituteofMathematicsandComputerScience,ZhejiangNormalUniversity,JinhuaZhejiang
Received:Oct.8
th
,2021;accepted:Oct.29
th
,2021;published:Nov.9
th
,2021
Abstract
BasedontheresearchofH-ToeplitzoperatorsonBergmanspace,thisarticlemainly
studiesthecommutantsofH-ToeplitzoperatorsonDirichletspace.
©ÙÚ^:oŠY.Dirichlet˜mþH-ToeplitzŽf†5[J].A^êÆ?Ð,2021,10(11):3699-3711.
DOI:10.12677/aam.2021.1011393
oŠY
Keywords
H-ToeplitzOperators,DirichletSpace,Commutant
Copyright
c
2021byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution InternationalLicense(CCBY4.0).
http://creativecommons.org/licenses/by/4.0/
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DOI:10.12677/aam.2021.10113933701A^êÆ?Ð
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DOI:10.12677/aam.2021.10113933702A^êÆ?Ð
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DOI:10.12677/aam.2021.10113933703A^êÆ?Ð
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DOI:10.12677/aam.2021.10113933704A^êÆ?Ð
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a
6
···
√
3a
4
√
3a
2
3
√
6
a
5
q
3
2
a
1
a
6
a
0
3
2
√
3
a
7
···
2a
5
2a
3
√
2a
6
√
2a
2
2
√
3
a
7
q
4
3
a
1
a
8
···
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.











DOI:10.12677/aam.2021.10113933705A^êÆ?Ð
oŠY
B
φ
Š‘Ý•
B
∗
φ
=

























a
2
√
2a
3
√
3a
4
2a
5
···
a
0
1
√
2
a
1
√
3a
2
2a
3
···
1
√
2
a
3
a
4
3
√
6
a
5
√
2a
6
···
1
√
2
b
1
a
0
q
3
2
a
1
√
2a
2
···
1
√
3
a
4
2
√
6
a
5
a
6
2
√
3
a
7
···
√
3b
2
q
3
2
b
1
a
0
q
4
3
a
1
···
.
.
.
.
.
.
.
.
.
.
.
.

























dB
φ
ÝŒB
φ
ÝLˆª¥2n+1(n=1,2,...)TT•HankelŽfÝLˆª,ó
ê•ToeplitzŽfÝLˆª.
e5·‚ò‰ÑäNDirichletH-ToeplitzÝ:
½Âéuφ(z) =
P
∞
i=1
a
i
z
i
+
P
∞
j=1
b
j
z
j
∈L
∞
(D),e˜‡Ý(C
m,n
)(m,n)g‘÷ve'X:
C
m,n
=

















q
m
j
a
m−j
n= 2jm≥j
q
j
m
b
j−m
n= 2j,m<j
m+j+1
√
m(j+1)
a
m+j+1
n+2j+1
Ù¥m,n,jÑ´ê,KŒòÕÝ(C
m,n
)½Â•˜‡DirichletH-ToeplitzÝ.qd(C
m,n
)
ÝŒ(C
m,n
)÷vC
1,2
= C
j,2j
,j≥1.
^B(D)L«D¥k.‚5Žf8Ü,e¡·‚y²éuNγ:G→B(D),e½Â•‡o
´{φ∈L
∞
(D) : φ´NÚ}‡o´{φ∈C
1
(D) : φ´NÚ},Kγ´˜˜.
·K2.5½Â•γ(φ) =B
φ
¼êγ: G→B(D)o´˜˜,Ù¥G‡o´˜m{φ∈L
∞
(D) :
φ´NÚ}‡o´˜m{φ∈C
1
(D) : φ´NÚ}
y²œ¹1-G={φ∈L
∞
(D):φ´NÚ},…φ,ψ∈L
∞
(D)´DþNڼꧽÂ
•φ(z)=
P
∞
i=1
a
i
z
i
+
P
∞
j=1
b
j
z
j
,ψ(z) =
P
∞
j=1
a
0
i
z
i
+
P
∞
j=1
b
0
j
z
j
,eB
φ
= B
ψ
,Kk(B
φ
−B
ψ
)(e
n
) =
0,n>0,AO/,(B
φ
−B
ψ
)(1) = B
φ−ψ
(1) = 0,|^½Â,
B
φ−ψ
(1) = PM
φ−ψ
K(1) = p(
∞
X
i=1
(a
i
−a
0
i
)z
i
+
∞
X
j=1
(b
j
−b
0
j
)z
j
) =
∞
X
i=1
(a
i
−a
0
i
)z
i
= 0
ÏdŒa
i
−a
0
i
= 0,i≥1,=a
i
= a
0
i
,i≥1
DOI:10.12677/aam.2021.10113933706A^êÆ?Ð
oŠY
…B
φ−ψ
(e
2
) = 0.K
B
φ−ψ
(e
2
) = PM
φ−ψ
K(e
2
) = PM
φ−ψ
e
1
= PM
φ−ψ
z
= P(
∞
X
i=1
(a
i
−a
0
i
)z
i
+
∞
X
j=1
(b
j
−b
0
j
)z
j
)z
= P
∞
X
i=1
(a
i
−a
0
i
)z
i+1
+
∞
X
j=1
(b
j
−b
0
j
)z
j
z)
=
1
2
(b
1
−b
0
1
) = 0
Kkb
1
−b
0
1
= 0,b
1
= b
0
1
.
X,B
φ−ψ
(e
4
) = 0,K
B
φ−ψ
(e
4
) = PM
φ−ψ
K(e
4
) = PM
φ−ψ
e
2
= PM
φ−ψ
1
√
2
z
2
= P
1
√
2
(
∞
X
i=1
(a
i
−a
0
i
)z
i+2
+
∞
X
j=1
(b
j
−b
0
j
)z
j
)z
2
= P
1
√
2
∞
X
j=1
(b
j
−b
0
j
)z
j
z
2
) =
1
√
2
(b
2
−b
0
2
)P(z
2
z
2
)
= 0
¤±7kb
2
−b
0
2
= 0,=b
2
= b
0
2
.
•g^aq•{òB
φ
Š^e
6
,e
8
,···þ,Œb
j
= b
0
j
,j≥1.Ïdφ= ψ,¤±γ´˜˜.
œ¹2-G={φ∈C
1
(
D):φ´NÚ},b½γ(φ)=0,=B
φ
=0,φ∈C
1
(D),Ké?¿
êm,n,k
0 =

B
φ
z
2m
,z
n

=
√
2mnhPM
φ
e
m
,e
n
i
=
√
2hT
φ
z
m
,z
n
i
=
√
2nhφz
m
,z
n
i
H
2
Ïdhφz
m
,z
n
i
H
2
=0,¤±φ3>.þ•0,=φ|
∂D
=0.∵φ´NÚ,∴ŒdÑtÈ©úªφ(z)=
1
2π
R
2π
0
(R
2
−r
2
)φ(Re
iθ
)
R
2
−2Rrcos(θ−ϕ)+r
2
dθ,z∈D.φ(Re
iθ
)=0ž,Kφ(z)=0,z∈D,=φ3DSð•0.γ´˜
˜.
3.H-To eplitzŽf†5
3ù˜Ü©·‚ïÄH-ToeplitzŽf†5,Ù¥H-ToeplitzŽfÎÒ•)ÛÚÝ)Û
,˜„/,ü‡ÎÒÑ•)Û…ÎÒgêØÓH-ToeplitzŽf´ØŒ±†,e¡~fŒ±
ƒ±y².
~f3.1-φ(z)=z,ψ(z)=z
3
,KB
z
(e
4
(z))=PM
z
K(e
4
(z))=PM
z
e
2
(z)=PM
z
1
√
2
z
2
=
DOI:10.12677/aam.2021.10113933707A^êÆ?Ð
oŠY
1
√
2
z
3
,B
z
3
(e
4
(z)) = PM
z
3
K(e
4
(z)) = PM
z
3
e
2
(z) = PM
z
3
1
√
2
z
2
=
1
√
2
z
5
,Ïd,
B
z
B
z
3
(e
4
(z)) = B
z
(
1
√
2
z
5
) =
1
√
2
PM
z
K(z
5
) =
1
√
2
PM
z
√
5e
3
= 0,
B
z
3
B
z
(e
4
(z)) = B
z
3
(
1
√
2
z
3
) =
1
√
2
PM
z
3
K(z
3
) =
√
2
√
3
PM
z
3
e
2
=
√
3
2
P(z
3
z
2
) =
√
3
2
z.
¤±B
z
B
z
3
6= B
z
3
B
z
.
ü‡H-ToeplitzŽfÎÒ©O•)ÛÚÝ)Ûž,~X-φ(z)=z,ψ(z)=z,
KB
z
B
z
6= B
z
B
z
,Ï•
B
z
B
z
(z
2
) = B
z
PM
z
√
2z=
√
2
2
z,
B
z
B
z
(z
2
) = B
z
PM
z
√
2z= PM
z
K(
√
2z
2
) = 1.
ÏLþ¡~f,·‚uyéuü‡Ñ´)ÛÎÒ…ÎÒgêØÓH-ToeplitzŽf±9ü‡
©O´)ÛÚÝ)ÛH-ToeplitzŽf˜„´ØU†,¤±e¡·‚^˜‡½n5•xü
‡H-ToeplitzŽf3Dirichlet˜m¥Œ±†^‡.
½n3.2-φ(z)=
P
∞
n=1
a
n
z
n
Úψ(z)=
P
∞
m=1
b
m
z
m
,Ù¥a
n
,b
m
,n,m≥1Ñ´šK~ê,
…φ(0) = 0,ψ(0) = 0,eB
φ
†B
ψ
Œ†…=φ= 0½ψ= 0.
yÄk¿©5´w,,eφ=0½ψ=0,KB
φ
B
ψ
=B
ψ
B
φ
.Ùg´7‡5,eB
φ
B
ψ
=
B
ψ
B
φ
,K˜½kB
φ
B
ψ
(1) = B
ψ
B
φ
(1),|^½Â,
B
φ
B
ψ
(1) = PM
φ
KPM
ψ
K(1) = PM
φ
KPM
ψ
(1) = 0,
B
ψ
B
φ
(1) = PM
ψ
KPM
φ
K(1) = PM
ψ
KPM
φ
(1)
= PM
ψ
KP(
∞
X
n=1
a
n
z
n
) = PM
ψ
(
∞
X
n=1
a
n
√
nK(e
n
))
n= 2k+1ž
B
ψ
B
φ
(1) = PM
ψ
(
∞
X
k=1
a
2k+1
√
2k+1K(e
2k+1
)) = 0
n= 2kž
B
ψ
B
φ
(1) = PM
ψ
KPM
φ
K(1) = PM
ψ
KPM
φ
(1)
= PM
ψ
(
∞
X
k=1
a
2k
√
2kK(e
2k
)) = P(
∞
X
m=1
b
m
z
m
)(
√
2
∞
X
k=1
a
2k
z
k
)
=
1
2
a
1
b
1
+
∞
X
t=1
√
2(
∞
X
j=1
a
j+t
b
j
)z
t
Ïd,
1
2
a
1
b
1
+
P
∞
t=1
√
2(
P
∞
j=1
a
j+t
b
j
)z
t
=0,dua
n
,b
m
,n,m≥1Ñ´šK,Ka
1
b
1
=
0,a
j+t
b
j
= 0,j≥1,t≥1.Ïdb
j
= 0,j≥1,½a
i+t
= 0,i,t≥1,=b
j
= 0,j≥1,½a
i
= 0,i≥1,ù
DOI:10.12677/aam.2021.10113933708A^êÆ?Ð
oŠY
L²φ= 0½ψ= 0.
|^T½n·‚•Œ±ïÄü‡H-ToeplitzŽfÎÒÑ´NÚžÿ†5.
½n3.3-φ(z)=
P
∞
i=1
a
i
z
i
+
P
∞
j=1
b
j
z
j
,…ψ(z)=
P
∞
m=1
c
m
z
m
+
P
∞
n=1
d
n
z
n
´L
∞,1
(D)¥
ü‡NÚ¼ê,…kφ(0)=0=ψ(0),Ù¥a
i
,b
j
,c
m
,d
n
,i,j,m,n≥1Ñ´š"Iþ,…b
½a
i
c
k−2i
≥c
i
a
k−2i
,a
i
c
2i−j
≥c
i
c
2i−j
±9a
l
d
l
≥c
l
b
l
,b
j
c
2j+h
≥d
j
a
2j+h
,Ù¥k,j,hþ•óê,l≥1.
Ke^‡d:
(1)B
φ
B
ψ
= B
ψ
B
φ
,
(2)φ†ψ´‚5ƒ'.
yeφ†ψ‚5ƒ',w,B
φ
†B
ψ
Œ†.eB
φ
†B
ψ
Œ†,K˜½kB
φ
B
ψ
(z
4
) = B
ψ
B
φ
(z
4
),
B
φ
B
ψ
(z
4
) = PM
φ
KPM
ψ
K(z
4
) = PM
φ
KPM
ψ
(2e
2
)
= PM
φ
KPM
ψ
√
2z
2
=
√
2PM
φ
KP[(
∞
X
m=1
c
m
z
m
+
∞
X
n=1
d
n
z
n
)z
2
]
=
√
2PM
φ
KP(
∞
X
m=1
c
m
z
m+2
+
∞
X
n=1
d
n
z
n
z
2
)
=
√
2PM
φ
K(
∞
X
m=1
c
m
z
m+2
+d
1
z+
1
3
d
2
)
=
√
2PM
φ
(
∞
X
m=1
c
m
√
2z
m+2
2
+
∞
X
m=1
c
m
r
2(m+2)
m+3
z
m+3
2
+d
1
z+
1
3
d
2
)
=
√
2P(
∞
X
i=1
a
i
z
i
+
∞
X
j=1
b
j
z
j
)(
∞
X
m=1
c
m
√
2z
m+2
2
+
∞
X
m=1
c
m
r
2(m+2)
m+3
z
m+3
2
+d
1
z+
1
3
d
2
)
=
√
2(
∞
X
i=1
a
i
∞
X
m=1
c
m
√
2z
m+2i+2
2
+
∞
X
i=1
a
i
∞
X
m=1
c
m
r
2(m+2)
m+3
z
2i−m−3
2
+
1
2
a
1
d
1
+
∞
X
i=2
a
i
d
1
z
i−1
+
1
3
d
2
∞
X
i=1
a
i
z
i
+
∞
X
j=1
b
j
∞
X
m=1
c
m
√
2z
m−2j+2
2
)
B
ψ
B
φ
(z
4
) = PM
ψ
KPM
φ
K(z
4
) = PM
ψ
KPM
φ
(2e
2
)
= PM
ψ
KPM
φ
√
2z
2
=
√
2PM
ψ
KP[(
∞
X
i=1
a
i
z
i
+
∞
X
j=1
b
j
z
j
)z
2
]
=
√
2PM
ψ
KP(
∞
X
i=1
a
i
z
i+2
+
∞
X
j=1
b
j
z
j
z
2
)
=
√
2PM
ψ
K(
∞
X
i=1
a
i
z
i+2
+b
1
z+
1
3
b
2
)
=
√
2PM
ψ
(
∞
X
i=1
a
i
√
i+2K(e
i+2
)+b
1
z+
1
3
b
2
)
=
√
2PM
ψ
(
∞
X
i=1
a
i
√
i+2e
i+2
2
+
∞
X
i=1
a
i
√
i+2e
i+3
2
+b
1
z+
1
3
b
2
)
DOI:10.12677/aam.2021.10113933709A^êÆ?Ð
oŠY
=
√
2P(
∞
X
m=1
c
m
z
m
+
∞
X
n=1
d
n
z
n
)(
∞
X
i=1
a
i
√
2z
i+2
2
+
∞
X
i=1
a
i
r
2(i+2)
i+3
z
i+3
2
+b
1
z+
1
3
b
2
)
=
√
2(
∞
X
m=1
c
m
∞
X
i=1
a
i
√
2z
2m+i+2
2
+
∞
X
m=1
c
m
∞
X
i=1
a
i
r
2(i+2)
i+3
z
2m−i−3
2
+
1
2
c
1
b
1
+
∞
X
m=2
c
m
b
1
z
m−1
+
1
3
b
2
∞
X
m=1
c
m
z
m
+
∞
X
n=1
d
n
∞
X
i=1
a
i
√
2z
i−2n+2
2
)
'ªü>z‘Xê,Œ
√
2(
∞
X
i=1
a
i
c
2i−5
r
2i−3
i−1
+
1
2
a
1
d
1
+a
2
d
1
+
1
3
a
1
d
2
+
√
2
∞
X
j=1
b
j
c
2j
)
=
√
2(
∞
X
i=1
c
m
a
2m−5
r
2m−3
m−1
+
1
2
c
1
b
1
+c
2
b
1
+
1
3
b
2
c
1
+
√
2
∞
X
n=1
d
n
c
2n
).
½ödu
a
2m−5
c
2m−5
=
a
m
c
m
,m≥1,
b
1
d
1
=
a
1
c
1
=
a
2
c
2
=
b
2
d
2
…
a
2n
c
2n
=
b
n
d
n
,n≥1,?
a
i
c
i
=
a
i+1
c
i+1
,i=
1,2,3,
b
1
d
1
=
b
2
d
2
.
'ªü>z
2
‘Xê,Œ
√
2(
∞
X
i=1
a
i
c
2i−7
r
2i−5
i−1
+
1
2
a
1
d
1
+a
3
d
1
+
1
3
d
2
a
2
+
√
2
∞
X
j=1
b
j
c
2j+2
)
=
√
2(
∞
X
m=1
c
m
a
2m−7
r
2m−5
m−2
+
1
2
c
1
b
1
+c
3
b
1
+
1
3
b
2
c
2
+
√
2
∞
X
n=1
d
n
c
2n+2
)
½ödu
a
2m−7
c
2m−7
=
a
m
c
m
,m≥1,
b
1
d
1
=
a
1
c
1
=
a
3
c
3
…
a
2n+2
c
2n+2
=
b
n
d
n
,n≥1,?
a
i
c
i
=
a
i+1
c
i+1
,i=
1,2,3,4,5,
b
j+1
d
j+1
=
b
j
d
j
,j= 1,2.
'ªü>z
3
‘Xê,Œ
√
2(
√
2
∞
X
i=1
a
i
c
4−2i
+
∞
X
i=1
a
i
c
2i−9
r
2i−7
i−3
+
1
2
a
1
d
1
+a
4
d
1
+
1
3
d
2
a
3
+
√
2
∞
X
j=1
b
j
c
2j+4
)
=
√
2(
∞
X
m=1
c
m
a
4−2m
√
2+
∞
X
m=1
c
m
a
2m−9
r
2m−7
m−3
+
1
2
c
1
b
1
+c
4
b
1
+
1
3
b
2
c
3
+
√
2
∞
X
n=1
d
n
c
2n+4
)
½ödu
a
2m−9
c
2m−9
=
a
m
c
m
=
a
4−2m
c
4−2m
,m≥1,
b
1
d
1
=
a
1
c
1
=
a
4
c
4
…
a
2n+4
c
2n+4
=
b
n
d
n
,n≥1,?
a
i
c
i
=
a
i+1
c
i+1
,i=
1,2,3,4,5,6,
b
j+1
d
j+1
=
b
j
d
j
,j= 1,2,3.
X•g'e,·‚Œ±
a
k
c
k
=
a
k+1
c
k+1
,k≥1,
b
j+1
d
j+1
=
b
j
d
j
,j≥1,du
b
1
d
1
=
a
1
c
1
=
a
2
c
2
=
b
2
d
2
,
¤±Œ
a
t
c
t
=
b
t
d
t
= λ,Ù¥λ=
b
1
d
1
,•L²φ= λψ.
DOI:10.12677/aam.2021.10113933710A^êÆ?Ð
oŠY
ë•©z
[1]Lee,Y.J.(2009)FiniteSumsofToeplitzProductsontheDirichletSpace.JournalofMathe-
maticalAnalysisandApplications,357,504-515.https://doi.org/10.1016/j.jmaa.2009.04.035
[2]Chen, Y.and Dieu,N.Q.(2010)ToeplitzandHankelOperators withL
∞,1
SymbolsonDirichlet
Space.JournalofMathematicalAnalysisandApplications,369,368-376.
[3]Zhao,L.(2008)CommutativityofToeplitzOperatorsontheHarmonicDirichletSpace.Journal
ofMathematicalAnalysisandApplications,339,1148-1160.
https://doi.org/10.1016/j.jmaa.2007.07.030
[4]Lu,Y.,Hu,Y.andLiu,L.(2015)CompactToeplitzOperatorsontheWeightedDirichlet
Space.ActaMathematicaSinica,31,35-43.https://doi.org/10.1007/s10114-015-3380-z
[5]Yu,T.(2010)ToeplitzOperatorsontheDirichletSpace.IntegralEquationsandOperator
Theory,67,163-170.https://doi.org/10.1007/s00020-010-1754-2
[6]Stroethoff,K.(1990)CompactHankelOperatorsontheBergmanSpace.IllinoisJournalof
Mathematics,34,159-174.https://doi.org/10.1215/ijm/1255988500
[7]Axler,S.,Conway,J.B.andMcDonald,G.(1982)ToeplitzOperatorsonBergmanSpaces.
CanadianJournalofMathematics,34,466-483.
https://doi.org/10.4153/CJM-1982-031-1
[8]Axler,S.,Cuckoric,andRao, N.V.(2000) Commutantsof AnalyticToeplitzOperatorson the
BergmanSpace.ProceedingsoftheAmericanMathematicalSociety,128,1951-1953.
https://doi.org/10.1090/S0002-9939-99-05436-2
[9]Gupta,A.andSingh,S.K.(2019)SlantH-ToeplitzOperatorsontheHardySpace.Journalof
theKoreanMathematicalSociety,56,703-721.
[10]Arora,S.C.andPaliwal,S.(2007)OnH-ToeplitzOperators.BulletinofPureandApplied
Mathematics,1,142-154.
DOI:10.12677/aam.2021.10113933711A^êÆ?Ð

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