Hilbert-Type Operators Acting on Bergman Spaces Hilbert-Type Operators Acting on Bergman Spaces
If μ is a positive Borel measure on the interval [0, 1) we let H μ be the Hankel matrix H μ = ( μ n , k ) n , k ≥ 0 with entries μ n , k = μ n + k , where, for n = 0 , 1 , 2 , … , μ n denotes the moment of order n of μ . This matrix formally induces an operator, called also H μ , on the space of all...
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| Vydáno v: | Computational methods and function theory Ročník 25; číslo 4; s. 863 - 888 |
|---|---|
| Hlavní autoři: | , , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
Berlin/Heidelberg
Springer Berlin Heidelberg
01.12.2025
Springer Nature B.V |
| Témata: | |
| ISSN: | 1617-9447, 2195-3724 |
| On-line přístup: | Získat plný text |
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| Abstract | If
μ
is a positive Borel measure on the interval [0, 1) we let
H
μ
be the Hankel matrix
H
μ
=
(
μ
n
,
k
)
n
,
k
≥
0
with entries
μ
n
,
k
=
μ
n
+
k
, where, for
n
=
0
,
1
,
2
,
…
,
μ
n
denotes the moment of order
n
of
μ
. This matrix formally induces an operator, called also
H
μ
, on the space of all analytic functions in the unit disc
D
as follows: If
f
is an analytic function in
D
,
f
(
z
)
=
∑
k
=
0
∞
a
k
z
k
,
z
∈
D
,
H
μ
(
f
)
is formally defined by
H
μ
(
f
)
(
z
)
=
∑
n
=
0
∞
∑
k
=
0
∞
μ
n
+
k
a
k
z
n
,
z
∈
D
.
This is a natural generalization of the classical Hilbert operator. This paper is devoted to studying the operators
H
μ
acting on the Bergman spaces
A
p
,
1
≤
p
<
∞
. Among other results, we give a complete characterization of those
μ
for which
H
μ
is bounded or compact on the space
A
p
when
p
is either 1 or greater than 2. We also give a number of results concerning the boundedness and compactness of
H
μ
on
A
p
for the other values of
p
, as well as on its membership in the Schatten classes
S
p
(
A
2
)
. |
|---|---|
| AbstractList | If
μ
is a positive Borel measure on the interval [0, 1) we let
H
μ
be the Hankel matrix
H
μ
=
(
μ
n
,
k
)
n
,
k
≥
0
with entries
μ
n
,
k
=
μ
n
+
k
, where, for
n
=
0
,
1
,
2
,
…
,
μ
n
denotes the moment of order
n
of
μ
. This matrix formally induces an operator, called also
H
μ
, on the space of all analytic functions in the unit disc
D
as follows: If
f
is an analytic function in
D
,
f
(
z
)
=
∑
k
=
0
∞
a
k
z
k
,
z
∈
D
,
H
μ
(
f
)
is formally defined by
H
μ
(
f
)
(
z
)
=
∑
n
=
0
∞
∑
k
=
0
∞
μ
n
+
k
a
k
z
n
,
z
∈
D
.
This is a natural generalization of the classical Hilbert operator. This paper is devoted to studying the operators
H
μ
acting on the Bergman spaces
A
p
,
1
≤
p
<
∞
. Among other results, we give a complete characterization of those
μ
for which
H
μ
is bounded or compact on the space
A
p
when
p
is either 1 or greater than 2. We also give a number of results concerning the boundedness and compactness of
H
μ
on
A
p
for the other values of
p
, as well as on its membership in the Schatten classes
S
p
(
A
2
)
. If μ is a positive Borel measure on the interval [0, 1) we let Hμ be the Hankel matrix Hμ=(μn,k)n,k≥0 with entries μn,k=μn+k, where, for n=0,1,2,…, μn denotes the moment of order n of μ. This matrix formally induces an operator, called also Hμ, on the space of all analytic functions in the unit disc D as follows: If f is an analytic function in D, f(z)=∑k=0∞akzk, z∈D, Hμ(f) is formally defined by Hμ(f)(z)=∑n=0∞∑k=0∞μn+kakzn,z∈D.This is a natural generalization of the classical Hilbert operator. This paper is devoted to studying the operators Hμ acting on the Bergman spaces Ap, 1≤p<∞. Among other results, we give a complete characterization of those μ for which Hμ is bounded or compact on the space Ap when p is either 1 or greater than 2. We also give a number of results concerning the boundedness and compactness of Hμ on Ap for the other values of p, as well as on its membership in the Schatten classes Sp(A2). |
| Author | Galanopoulos, Petros Girela, Daniel Aguilar-Hernández, Tanausú |
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| Cites_doi | 10.1016/j.jfa.2008.02.009 10.1090/S0002-9904-1969-12181-6 10.1090/S0002-9939-1975-0374886-9 10.1215/S0012-7094-77-04409-X 10.1007/s00365-012-9157-z 10.1090/S0002-9947-1966-0187099-X 10.1307/mmj/1029004756 10.1090/surv/138 10.4064/sm200-3-1 10.4153/CJM-1986-043-4 10.2307/1970375 10.4064/sm-140-2-191-198 10.1016/j.jmaa.2017.04.002 10.1016/j.jmaa.2013.11.046 10.1007/978-1-4612-0497-8 10.1007/s13163-018-0288-z 10.1007/s00025-023-01887-6 10.5186/aasfm.2012.3715 10.1112/plms/s3-63.3.595 10.1090/S0002-9947-03-03354-3 |
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| Keywords | Compact operator Schatten classes The Hilbert matrix 47B35 Bounded operator Generalized Hilbert operator Duality Bergman spaces Carleson measures 30H20 |
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| References | R Zhao (560_CR24) 2003; 69 DH Luecking (560_CR19) 1991; 63 D Girela (560_CR12) 2019; 32 WW Hastings (560_CR14) 1975; 52 E Diamantopoulos (560_CR5) 2004; 48 P Galanopoulos (560_CR11) 2010; 200 L Carleson (560_CR3) 1962; 76 PL Duren (560_CR10) 2004 560_CR13 H Hedenmalm (560_CR15) 2000 B Lanucha (560_CR18) 2012; 37 DH Luecking (560_CR20) 1993; 40 JM Anderson (560_CR2) 1974; 270 A Aleman (560_CR1) 2012; 36 Ch Chatzifountas (560_CR4) 2014; 413 C Horowitz (560_CR16) 1977; 44 560_CR25 M Jevtić (560_CR17) 2017; 453 BD MacCluer (560_CR21) 1986; 38 M Dostanić (560_CR6) 2008; 254 E Diamantopoulos (560_CR7) 2000; 140 560_CR9 M Tjani (560_CR22) 2003; 355 H Widom (560_CR23) 1966; 121 PL Duren (560_CR8) 1969; 75 |
| References_xml | – volume: 254 start-page: 2800 year: 2008 ident: 560_CR6 publication-title: J. Funct. Anal. doi: 10.1016/j.jfa.2008.02.009 – volume: 75 start-page: 143 year: 1969 ident: 560_CR8 publication-title: Bull. Am. Math. Soc. doi: 10.1090/S0002-9904-1969-12181-6 – volume: 69 start-page: 605 issue: 3–4 year: 2003 ident: 560_CR24 publication-title: Acta Sci. Math. (Szeged) – volume: 52 start-page: 237 year: 1975 ident: 560_CR14 publication-title: Proc. Am. Math. Soc. doi: 10.1090/S0002-9939-1975-0374886-9 – volume-title: Bergman Spaces, Mathematical Surveys and Monographs year: 2004 ident: 560_CR10 – volume: 48 start-page: 1067 issue: 3 year: 2004 ident: 560_CR5 publication-title: Ill. J. Math. – volume: 44 start-page: 201 issue: 1 year: 1977 ident: 560_CR16 publication-title: Duke Math. J. doi: 10.1215/S0012-7094-77-04409-X – volume: 36 start-page: 353 issue: 3 year: 2012 ident: 560_CR1 publication-title: Const. Approx. doi: 10.1007/s00365-012-9157-z – volume: 121 start-page: 1 year: 1966 ident: 560_CR23 publication-title: Trans. Am. Math. Soc. doi: 10.1090/S0002-9947-1966-0187099-X – volume: 40 start-page: 333 issue: 2 year: 1993 ident: 560_CR20 publication-title: Mich. Math. J. doi: 10.1307/mmj/1029004756 – ident: 560_CR25 doi: 10.1090/surv/138 – volume: 200 start-page: 201 issue: 3 year: 2010 ident: 560_CR11 publication-title: Studia Math. doi: 10.4064/sm200-3-1 – volume: 38 start-page: 878 issue: 4 year: 1986 ident: 560_CR21 publication-title: Can. J. Math. doi: 10.4153/CJM-1986-043-4 – volume: 76 start-page: 547 year: 1962 ident: 560_CR3 publication-title: Ann. Math. doi: 10.2307/1970375 – volume: 140 start-page: 191 issue: 2 year: 2000 ident: 560_CR7 publication-title: Studia Math. doi: 10.4064/sm-140-2-191-198 – volume: 453 start-page: 241 issue: 1 year: 2017 ident: 560_CR17 publication-title: J. Math. Anal. Appl. doi: 10.1016/j.jmaa.2017.04.002 – volume: 413 start-page: 154 issue: 1 year: 2014 ident: 560_CR4 publication-title: J. Math. Anal. Appl. doi: 10.1016/j.jmaa.2013.11.046 – volume-title: Theory of Bergman Spaces, Graduate Texts in Mathematics 199 year: 2000 ident: 560_CR15 doi: 10.1007/978-1-4612-0497-8 – volume: 32 start-page: 799 issue: 3 year: 2019 ident: 560_CR12 publication-title: Rev. Mat. Complut. doi: 10.1007/s13163-018-0288-z – ident: 560_CR13 doi: 10.1007/s00025-023-01887-6 – volume: 37 start-page: 161 year: 2012 ident: 560_CR18 publication-title: Ann. Acad. Sci. Fenn. Math. doi: 10.5186/aasfm.2012.3715 – volume: 63 start-page: 595 issue: 3 year: 1991 ident: 560_CR19 publication-title: Proc. Lond. Math. Soc. doi: 10.1112/plms/s3-63.3.595 – volume: 270 start-page: 12 year: 1974 ident: 560_CR2 publication-title: J. Reine Angew. Math. – volume: 355 start-page: 4683 issue: 11 year: 2003 ident: 560_CR22 publication-title: Trans. Am. Math. Soc. doi: 10.1090/S0002-9947-03-03354-3 – ident: 560_CR9 |
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| Snippet | If
μ
is a positive Borel measure on the interval [0, 1) we let
H
μ
be the Hankel matrix
H
μ
=
(
μ
n
,
k
)
n
,
k
≥
0
with entries
μ
n
,
k
=
μ
n
+
k
, where, for... If μ is a positive Borel measure on the interval [0, 1) we let Hμ be the Hankel matrix Hμ=(μn,k)n,k≥0 with entries μn,k=μn+k, where, for n=0,1,2,…, μn denotes... |
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| StartPage | 863 |
| SubjectTerms | Analysis Analytic functions Computational Mathematics and Numerical Analysis Functions of a Complex Variable Hankel matrices Mathematical analysis Mathematical functions Mathematics Mathematics and Statistics Operators (mathematics) |
| Subtitle | Hilbert-Type Operators Acting on Bergman Spaces |
| Title | Hilbert-Type Operators Acting on Bergman Spaces |
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