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AdvancesinAppliedMathematicsA^êÆ?Ð,2021,10(12),4489-4497
PublishedOnlineDecember2021inHans.http://www.hanspub.org/journal/aam
https://doi.org/10.12677/aam.2021.1012478
H-ToeplitzŽf“ê5Ÿ
ùùù777777
úô“‰ŒÆ,êƆOŽÅ‰ÆÆ,úô7u
ÂvFϵ2021c1127F¶¹^Fϵ2021c1223F¶uÙFϵ2021c1230F
Á‡
©Ì‡ïÄBergman˜mþH-ToeplitzŽf“ê5Ÿ"1˜Ù0ƒ'ïÄµ!
ÄVg9˜̇(J"1Ù‰Ñ©Ì‡(Jy²§y²[àgÎÒH-Toeplitz
ŽfEé¡5"
'…c
Bergman˜m§H-ToeplitzŽf§Eé¡5
AlgebraicPropertiesofH-Toeplitz
Operator
JinjinLiang
CollegeofMathematicsandComputerScience,ZhejiangNormalUniversity,JinhuaZhejiang
Received:Nov.27
th
,2021;accepted:Dec.23
rd
,2021;published:Dec.30
th
,2021
Abstract
Inthispaper,wemainlystudythealgebraicpropertiesofH-toeplitzoperatorson
BergmanSpaces.InChapter1,weintroducetherelatedresearchbackground,basic
©ÙÚ^:ù77.H-ToeplitzŽf“ê5Ÿ[J].A^êÆ?Ð,2021,10(12):4489-4497.
DOI:10.12677/aam.2021.1012478
ù77
conceptsandsomemainresults.InChapter2,theproofofthemainresultsofthis
paperisgiven,andthecomplexsymmetryofquasihomogeneoussignedH-toeplitz
operatorsisproved.
Keywords
BergmanSpace,H-ToeplitzOperator,ComplexSymmetry
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.2021.10124784491A^êÆ?Ð
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DOI:10.12677/aam.2021.10124784492A^êÆ?Ð
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a
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b
1
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a
1
1
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a
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a
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a
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1
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a
5
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3
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1
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(D,dA)þEÝŽfC:L
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2
=I
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2
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DOI:10.12677/aam.2021.10124784493A^êÆ?Ð
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φ
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i= he
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i
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∞
X
j=0
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j
z
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i+
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z
n
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m
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m
,
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i
z
i
z
n
i
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m
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i+n
i
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n+1
1
m+1
a
m−n
=
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n+1
m+1
a
m−n
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hCB
φ
Ce
2n
,e
m
i=
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m+1
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n+1hz
m
,
∞
X
j=0
b
j
z
j
z
n
i
=
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m+1
√
n+1
∞
X
j=0
b
j
hz
m+j
,z
n
i
=
√
m+1
√
n+1
1
n+1
b
n−m
=
r
m+1
n+1
b
n−m
.
DOI:10.12677/aam.2021.10124784494A^êÆ?Ð
ù77
hCB
φ
Ce
2n+1
,e
m
i= hCe
m
,B
φ
Ce
2n+1
i
= he
m
,B
φ
e
2n+1
i= he
m
,P
L
2
a
M
φ
K(e
2n+1
)i
= he
m
,P
L
2
a
M
φ
e
n+1
i= he
m
,M
φ
e
n+1
i= he
m
,φe
n+1
i
= h
√
m+1z
m
,(
∞
X
i=0
a
i
z
i
+
∞
X
j=0
b
j
z
j
)
√
n+2z
n+1
i
=
√
m+1
√
n+1hz
m
,
∞
X
i=0
a
i
z
i
z
n+1
i+hz
m
,
∞
X
j=0
b
j
z
j
¯z
n+1
i
=
√
m+1
√
n+1
∞
X
i=0
a
i
hz
m
,z
i
z
n+1
i
=
√
m+1
√
n+2
n+m+2
a
m+n+1
.
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φ
CÝ/ªXe:
CB
φ
C=












a
0
1
√
2
a
1
1
√
2
b
1
1
√
3
a
2
1
√
3
b
2
1
2
a
3
1
2
b
3
···
1
√
2
a
1
2
3
a
2
a
0
√
6
4
a
3
1
√
2
b
1
2
√
2
√
5
a
4
1
√
2
b
2
···
1
√
3
a
2
√
6
4
a
3
q
2
3
a
1
3
5
a
4
a
0
1
√
3
a
5
√
3
2
b
1
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DOI:10.12677/aam.2021.10124784495A^êÆ?Ð
ù77
n>mž,
r
m+1
n+1
a
n−m
=
r
m+1
n+1
b
n−m
;
r
m+1
n+1
a
n−m
=
√
m+1
√
m+2
m+n+2
a
m+n+1
;
r
m+1
n+1
b
n−m
=
√
m+1
√
n+2
m+n+2
a
m+n+1
.
n≤mž,
r
n+1
m+1
a
m−n
=
r
n+1
m+1
b
m−n
;
r
n+1
m+1
a
m−n
=
√
m+1
√
n+2
n+m+2
a
m+n+1
;
r
n+1
m+1
b
m−n
=
√
m+1
√
n+2
n+m+2
a
m+n+1
.
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1
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2
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3
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m+n+1
→∞.a
1
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1
=
a
2
=b
2
=a
3
=b
3
=···=a
n−m
=b
n−m
=a
m+n+1
→∞,m,n→∞.Ý¥z‡ƒÑ•
¹,qÏ•3L
2
(D)¥Ý÷vsup
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2
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j=0
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2
<∞,¤±m,n→∞,
lim
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P
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i
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2
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j
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=0,ù¿›Xéu¤kiÚj,a
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ë•©z
[1]Ahern,P.and
ˇ
Cuˇckovi´e,
ˇ
Z.(2001)ATheoremofBrown-HalmosTypeforBergmanSpace
ToeplitzOperators.JournalofFunctionalAnalysis,187,200-210.
https://doi.org/10.1006/jfan.2001.3811
[2]Brown,A.andHalmos,P.R.(1964)AlgebraicPropertiesofToeplitzOperators.Journalf¨ur
dieReineundAngewandteMathematik,213,89-102. https://doi.org/10.1515/crll.1964.213.89
[3]Louhichi, I. and Zakariasy, L.(2005) On Toeplitz Operatorswith Quasihomogeneous Symbols.
ArchivderMathematik(Basel),85,248-257.https://doi.org/10.1007/s00013-005-1198-0
[4]Gupta,A.andSingh,S.K.(2021)H-ToeplitzOperatorsonBergmanSpace.Bulletinofthe
KoreanMathematicalSociety,58,327-347.
[5]Lu,Y.andZhang,B.(2011)CommutingHankelOperatorandToeplitzOperatoronthe
BergmanSpace.ChineseAnnalsofMathematics,SeriesA,32,519-530.
[6]Chen,Y.,Lee,Y.J.andZhao,Y.(2021)ComplexSymmetryofToeplitzOperators.Preprint.
DOI:10.12677/aam.2021.10124784496A^êÆ?Ð
ù77
[7]Garcia,S.R.andPutinar,M.(2006)ComplexSymmetricOperatorsandApplications.Trans-
actionsoftheAmericanMathematicalSociety,358,1285-1315.
https://doi.org/10.1090/S0002-9947-05-03742-6
DOI:10.12677/aam.2021.10124784497A^êÆ?Ð

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