| Title:
|
The quasi-canonical solution operator to $\bar{\partial}$ restricted to the Fock-space (English) |
| Author:
|
Schneider, Georg |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
55 |
| Issue:
|
4 |
| Year:
|
2005 |
| Pages:
|
947-956 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We consider the solution operator $S\:\mathcal F_{\mu ,(p,q)}\rightarrow L^2(\mu )_{(p,q)}$ to the $\bar{\partial }$-operator restricted to forms with coefficients in $\mathcal F_{\mu }= \bigl \lbrace f\: f \text{is} \text{entire} \text{and} \int _{\mathbb{C}^n} |f(z)|^2\mathrm{d}\mu (z) <\infty \bigr \rbrace $. Here $\mathcal F_{\mu ,(p,q)}$ denotes $(p,q)$-forms with coefficients in $\mathcal F_{\mu }$, $L^2(\mu )$ is the corresponding $L^2$-space and $\mu $ is a suitable rotation-invariant absolutely continuous finite measure. We will develop a general solution formula $S$ to $\bar{\partial }$. This solution operator will have the property $Sv\bot \mathcal F_{(p,q)}\, \forall \,v \in \mathcal F_{(p,q+1)}$. As an application of the solution formula we will be able to characterize compactness of the solution operator in terms of compactness of commutators of Toeplitz-operators $[T_{\bar{z_i}},T_{z_i}]= [T^*_{{z_i}},T_{z_i}]\:\mathcal F_\mu \rightarrow L^2(\mu )$. (English) |
| Keyword:
|
Fock-space |
| Keyword:
|
Hankel-operator |
| Keyword:
|
reproducing kernel |
| MSC:
|
32A15 |
| MSC:
|
32W05 |
| MSC:
|
35N15 |
| MSC:
|
47B35 |
| idZBL:
|
Zbl 1081.47035 |
| idMR:
|
MR2184376 |
| . |
| Date available:
|
2009-09-24T11:29:20Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/128037 |
| . |
| Reference:
|
[1] V. Bargmann: On a Hilbert Space of analytic functions and an associated integral transform.Commun. Pure Appl. Math. 14 (1961), 187–214. Zbl 0107.09102, MR 0157250, 10.1002/cpa.3160140303 |
| Reference:
|
[2] S. Fu and E. Straube: Compactness in the $\bar{\partial }$-Neumann problem.In: Proc. conf. Complex analysis and geometry, Ohio, Ohio State Univ. Math. Res Inst. Publ., 2001, pp. 141–160. MR 1912737 |
| Reference:
|
[3] L. Hörmander: $L^2$-estimates and existence theorems for the $\bar{\partial }$ operator.Acta Math. 113 (1965), 89–152. MR 0179443, 10.1007/BF02391775 |
| Reference:
|
[4] S. Axler: The Bergman space, the Bloch space, and commutators of multiplikation-operators.Duke Math. J. 53 (1986), 315–332. MR 0850538 |
| Reference:
|
[5] J. Arazy, S. Fischer and J. Peetre: Hankel-operators on weighted Bergman spaces.Amer. J. Math. 110 (1988), 989–1053. MR 0970119, 10.2307/2374685 |
| Reference:
|
[6] F. F. Bonsall: Hankel-operators on the Bergman space for the disc.J. London Math. Soc. 33 (1986), 355–364. Zbl 0604.47014, MR 0838646, 10.1112/jlms/s2-33.2.355 |
| Reference:
|
[7] S. Janson: Hankel-operators between weighted Bergman spaces.Ark. Math. 26 (1988), 205–219. MR 1050105, 10.1007/BF02386120 |
| Reference:
|
[8] R. Rochberg: Trace ideal criteria for Hankel-operators and commutators.Indiana Univ. Math. J. 31 (1982), 913–925. Zbl 0514.47020, MR 0674875, 10.1512/iumj.1982.31.31062 |
| Reference:
|
[9] R. Wallsten: Hankel-operators between weighted Bergman-spaces in the ball.Ark. Math. 28 (1990), 183–192. Zbl 0705.47023, MR 1049650, 10.1007/BF02387374 |
| Reference:
|
[10] K. H. Zhu: Hilbert-Schmidt Hankel-operators on the Bergman space.Proc. Amer. Math. Soc. 109 (1990), 721–730. Zbl 0731.47028, MR 1013987, 10.1090/S0002-9939-1990-1013987-7 |
| Reference:
|
[11] F. Haslinger: The canonical solution operator to $\bar{\partial }$ restricted to spaces of entire functions.Ann. Fac. Sci. Toulouse Math. 11 (2002), 57–70. MR 1986383, 10.5802/afst.1018 |
| Reference:
|
[12] F. Haslinger: The canonical solution operator to $\bar{\partial }$ restricted to Bergman-spaces.Proc. Amer. Math. Soc. 129 (2001), 3321–3329. MR 1845009, 10.1090/S0002-9939-01-05953-6 |
| Reference:
|
[13] L. Hörmander: An Introduction to Complex Analysis in Several Variables.Von Nostand, Princeton, 1966. MR 0203075 |
| Reference:
|
[14] W. Knirsch: Kompaktheit des $\bar{\partial }$-Neumann Operators.Dissertation, Universität Wien, Wien, 2000. |
| Reference:
|
[15] N. Salinas, A. Sheu and H. Upmeier: Toeplitz-operators on pseudoconvex domains and foliation $C^*$-algebras.Ann. Math. 130 (1989), 531–565. MR 1025166, 10.2307/1971454 |
| Reference:
|
[16] G. Schneider: Hankel-operators with anti-holomorphic symbols on the Fock-space.Proc. Amer. Math. Soc. 132 (2004), 2399–2409. MR 2052418, 10.1090/S0002-9939-04-07362-9 |
| Reference:
|
[17] G. Schneider: Non-compactness of the solution operator to $\bar{\partial }$ on the Fock-space in several dimensions.Math. Nachr. 278 (2005), 312–317. MR 2110534, 10.1002/mana.200310242 |
| Reference:
|
[18] S. Krantz: Compactness of the $\bar{\partial }$-Neumann operator.Proc. Amer. Math. Soc. 103 (1988), 1136–1138. Zbl 0736.35071, MR 0954995, 10.1090/S0002-9939-1988-0954995-2 |
| Reference:
|
[19] S. Fu and E. Straube: Compactness of the $\bar{\partial }$-Neumann problem on convex domains.J. Functional Analysis 159 (1998), 629–641. MR 1659575, 10.1006/jfan.1998.3317 |
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