Some Closed Range Integral Operators on Spaces of Analytic Functions
Our main result is a characterization of g for which the operator is bounded below on the Bloch space. We point out analogous results for the Hardy space H 2 and the Bergman spaces A p for 1 ≤ p < ∞. We also show the companion operator is never bounded below on H 2 , Bloch, nor BMOA , but may be...
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| Published in | Integral equations and operator theory Vol. 69; no. 1; pp. 87 - 99 |
|---|---|
| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
Basel
SP Birkhäuser Verlag Basel
01.01.2011
|
| Subjects | |
| Online Access | Get full text |
| ISSN | 0378-620X 1420-8989 |
| DOI | 10.1007/s00020-010-1827-2 |
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| Abstract | Our main result is a characterization of
g
for which the operator
is bounded below on the Bloch space. We point out analogous results for the Hardy space
H
2
and the Bergman spaces
A
p
for 1 ≤
p
< ∞. We also show the companion operator
is never bounded below on
H
2
, Bloch, nor
BMOA
, but may be bounded below on
A
p
. |
|---|---|
| AbstractList | Our main result is a characterization of
g
for which the operator
is bounded below on the Bloch space. We point out analogous results for the Hardy space
H
2
and the Bergman spaces
A
p
for 1 ≤
p
< ∞. We also show the companion operator
is never bounded below on
H
2
, Bloch, nor
BMOA
, but may be bounded below on
A
p
. |
| Author | Anderson, Austin |
| Author_xml | – sequence: 1 givenname: Austin surname: Anderson fullname: Anderson, Austin email: austina@hawaii.edu organization: Department of Mathematics, University of Hawaii |
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| Cites_doi | 10.2307/2373807 10.1007/BF02788078 10.1512/iumj.1979.28.28042 10.1007/BF01445229 10.1512/iumj.1997.46.1373 10.1007/s10114-008-7088-1 10.1007/3-7643-7314-8_9 10.1090/surv/138 10.4171/RMI/501 10.1090/conm/232/03406 10.1090/S0002-9939-1987-0875398-4 |
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| Keywords | BMOA multiplication operator Bergman Hardy integral operator 30D45 closed range Secondary 47G10 Volterra operator Cesaro operator bounded below Primary 47B38 Bloch |
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| References | Cowen, MacCluer (CR4) 1995 Garnett (CR7) 2007 Duren, Romberg, Shields (CR6) 1969; 238 Aleman, Cima (CR1) 2001; 85 McDonald, Sundberg (CR9) 1979; 28 Ramey, Ullrich (CR10) 1991; 291 CR8 Zhang, Chen (CR13) 2009; 25 CR14 Bourdon (CR3) 1987; 99 CR11 Aleman, Siskakis (CR2) 1997; 46 Dostanić (CR5) 2007; 23 Stegenga (CR12) 1976; 98 D.A. Stegenga (1827_CR12) 1976; 98 1827_CR14 C. Cowen (1827_CR4) 1995 1827_CR11 1827_CR8 G. McDonald (1827_CR9) 1979; 28 A. Aleman (1827_CR1) 2001; 85 M. Zhang (1827_CR13) 2009; 25 J.B. Garnett (1827_CR7) 2007 W. Ramey (1827_CR10) 1991; 291 P.L. Duren (1827_CR6) 1969; 238 M.R. Dostanić (1827_CR5) 2007; 23 A. Aleman (1827_CR2) 1997; 46 P.S. Bourdon (1827_CR3) 1987; 99 |
| References_xml | – volume: 23 start-page: 421 issue: 2 year: 2007 end-page: 436 ident: CR5 article-title: Integration operators on Bergman spaces with exponential weight publication-title: Rev. Mat. Iberoam. – volume: 98 start-page: 573 issue: 3 year: 1976 end-page: 589 ident: CR12 article-title: Bounded Toeplitz operators on and applications of the duality between and the functions of bounded mean oscillation publication-title: Am. J. Math. doi: 10.2307/2373807 – volume: 85 start-page: 157 year: 2001 end-page: 176 ident: CR1 article-title: An integral operator on and Hardy’s inequality publication-title: J. Anal. Math. doi: 10.1007/BF02788078 – year: 2007 ident: CR7 publication-title: Bounded Analytic Functions. Revised First Edition – ident: CR14 – volume: 28 start-page: 595 issue: 4 year: 1979 end-page: 611 ident: CR9 article-title: Toeplitz operators on the disc publication-title: Indiana Univ. Math. J. doi: 10.1512/iumj.1979.28.28042 – ident: CR11 – volume: 238 start-page: 32 year: 1969 end-page: 60 ident: CR6 article-title: Linear functionals on spaces with 0 < < 1 publication-title: J. Reine Angew. Math. – volume: 291 start-page: 591 issue: 4 year: 1991 end-page: 606 ident: CR10 article-title: Bounded mean oscillation of Bloch pull-backs publication-title: Math. Ann. doi: 10.1007/BF01445229 – volume: 46 start-page: 337 issue: 2 year: 1997 end-page: 356 ident: CR2 article-title: Integration operators on Bergman spaces publication-title: Indiana Univ. Math. J. doi: 10.1512/iumj.1997.46.1373 – ident: CR8 – volume: 25 start-page: 265 issue: 2 year: 2009 end-page: 278 ident: CR13 article-title: Weighted composition operators of into -Bloch spaces on the unit ball publication-title: Acta Math. Sin. (Engl. Ser.) doi: 10.1007/s10114-008-7088-1 – year: 1995 ident: CR4 publication-title: Composition Operators on Spaces of Analytic Functions – volume: 99 start-page: 563 issue: 3 year: 1987 end-page: 567 ident: CR3 article-title: Similarity of parts to the whole for certain multiplication operators publication-title: Proc. Am. Math. Soc. – volume-title: Bounded Analytic Functions. Revised First Edition year: 2007 ident: 1827_CR7 – volume-title: Composition Operators on Spaces of Analytic Functions year: 1995 ident: 1827_CR4 – volume: 28 start-page: 595 issue: 4 year: 1979 ident: 1827_CR9 publication-title: Indiana Univ. Math. J. doi: 10.1512/iumj.1979.28.28042 – volume: 291 start-page: 591 issue: 4 year: 1991 ident: 1827_CR10 publication-title: Math. Ann. doi: 10.1007/BF01445229 – volume: 46 start-page: 337 issue: 2 year: 1997 ident: 1827_CR2 publication-title: Indiana Univ. Math. J. doi: 10.1512/iumj.1997.46.1373 – volume: 98 start-page: 573 issue: 3 year: 1976 ident: 1827_CR12 publication-title: Am. J. Math. doi: 10.2307/2373807 – volume: 25 start-page: 265 issue: 2 year: 2009 ident: 1827_CR13 publication-title: Acta Math. Sin. (Engl. Ser.) doi: 10.1007/s10114-008-7088-1 – ident: 1827_CR8 doi: 10.1007/3-7643-7314-8_9 – volume: 85 start-page: 157 year: 2001 ident: 1827_CR1 publication-title: J. Anal. Math. doi: 10.1007/BF02788078 – volume: 238 start-page: 32 year: 1969 ident: 1827_CR6 publication-title: J. Reine Angew. Math. – ident: 1827_CR14 doi: 10.1090/surv/138 – volume: 23 start-page: 421 issue: 2 year: 2007 ident: 1827_CR5 publication-title: Rev. Mat. Iberoam. doi: 10.4171/RMI/501 – ident: 1827_CR11 doi: 10.1090/conm/232/03406 – volume: 99 start-page: 563 issue: 3 year: 1987 ident: 1827_CR3 publication-title: Proc. Am. Math. Soc. doi: 10.1090/S0002-9939-1987-0875398-4 |
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| Snippet | Our main result is a characterization of
g
for which the operator
is bounded below on the Bloch space. We point out analogous results for the Hardy space
H
2... |
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| Title | Some Closed Range Integral Operators on Spaces of Analytic Functions |
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