Article

Full entry | PDF   (0.3 MB)
Keywords:
Fock-space; Hankel-operator; reproducing kernel
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 $[T_{\bar{z_i}},T_{z_i}]= [T^*_{{z_i}},T_{z_i}]\:\mathcal F_\mu \rightarrow L^2(\mu )$.
References:
[1] V.  Bargmann: On a Hilbert Space of analytic functions and an associated integral transform. Commun. Pure Appl. Math. 14 (1961), 187–214. DOI 10.1002/cpa.3160140303 | MR 0157250 | Zbl 0107.09102
[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
[3] L.  Hörmander: $L^2$-estimates and existence theorems for the $\bar{\partial }$  operator. Acta Math. 113 (1965), 89–152. DOI 10.1007/BF02391775 | MR 0179443
[4] S.  Axler: The Bergman space, the Bloch space, and commutators of multiplikation-operators. Duke Math.  J. 53 (1986), 315–332. MR 0850538
[5] J.  Arazy, S.  Fischer and J.  Peetre: Hankel-operators on weighted Bergman spaces. Amer. J.  Math. 110 (1988), 989–1053. DOI 10.2307/2374685 | MR 0970119
[6] F. F.  Bonsall: Hankel-operators on the Bergman space for the disc. J.  London Math. Soc. 33 (1986), 355–364. DOI 10.1112/jlms/s2-33.2.355 | MR 0838646 | Zbl 0604.47014
[7] S.  Janson: Hankel-operators between weighted Bergman spaces. Ark. Math. 26 (1988), 205–219. DOI 10.1007/BF02386120 | MR 1050105
[8] R.  Rochberg: Trace ideal criteria for Hankel-operators and commutators. Indiana Univ. Math.  J. 31 (1982), 913–925. DOI 10.1512/iumj.1982.31.31062 | MR 0674875 | Zbl 0514.47020
[9] R.  Wallsten: Hankel-operators between weighted Bergman-spaces in the ball. Ark. Math. 28 (1990), 183–192. DOI 10.1007/BF02387374 | MR 1049650 | Zbl 0705.47023
[10] K. H.  Zhu: Hilbert-Schmidt Hankel-operators on the Bergman space. Proc. Amer. Math. Soc. 109 (1990), 721–730. DOI 10.1090/S0002-9939-1990-1013987-7 | MR 1013987 | Zbl 0731.47028
[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. DOI 10.5802/afst.1018 | MR 1986383
[12] F.  Haslinger: The canonical solution operator to  $\bar{\partial }$ restricted to Bergman-spaces. Proc. Amer. Math. Soc. 129 (2001), 3321–3329. DOI 10.1090/S0002-9939-01-05953-6 | MR 1845009
[13] L.  Hörmander: An Introduction to Complex Analysis in Several Variables. Von Nostand, Princeton, 1966. MR 0203075
[14] W.  Knirsch: Kompaktheit des $\bar{\partial }$-Neumann Operators. Dissertation, Universität Wien, Wien, 2000.
[15] N.  Salinas, A.  Sheu and H.  Upmeier: Toeplitz-operators on pseudoconvex domains and foliation $C^*$-algebras. Ann. Math. 130 (1989), 531–565. DOI 10.2307/1971454 | MR 1025166
[16] G.  Schneider: Hankel-operators with anti-holomorphic symbols on the Fock-space. Proc. Amer. Math. Soc. 132 (2004), 2399–2409. DOI 10.1090/S0002-9939-04-07362-9 | MR 2052418
[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. DOI 10.1002/mana.200310242 | MR 2110534
[18] S.  Krantz: Compactness of the $\bar{\partial }$-Neumann operator. Proc. Amer. Math. Soc. 103 (1988), 1136–1138. DOI 10.1090/S0002-9939-1988-0954995-2 | MR 0954995 | Zbl 0736.35071
[19] S.  Fu and E.  Straube: Compactness of the $\bar{\partial }$-Neumann problem on convex domains. J.  Functional Analysis 159 (1998), 629–641. DOI 10.1006/jfan.1998.3317 | MR 1659575

Partner of