Hankel operators with antiholomorphic symbols on the Fock space
HTML articles powered by AMS MathViewer
- by Georg Schneider PDF
- Proc. Amer. Math. Soc. 132 (2004), 2399-2409 Request permission
Abstract:
We consider Hankel operators of the form $H_{\overline {z}^k}: \mathcal {F}^m:=\{f : f \mbox { is entire and} \int _{\mathbb {C}^n}|f(z)|^2e^{-|z|^m}<\infty \}\rightarrow L^2(e^{-|z|^m})$. Here $k,m,n \in \mathbb {N}$. We show that in the case of one complex dimension the Hankel operators are compact but not Hilbert-Schmidt if $m>2k$.References
- J. Arazy, S. D. Fisher, and J. Peetre, Hankel operators on weighted Bergman spaces, Amer. J. Math. 110 (1988), no. 6, 989–1053. MR 970119, DOI 10.2307/2374685
- Sheldon Axler, The Bergman space, the Bloch space, and commutators of multiplication operators, Duke Math. J. 53 (1986), no. 2, 315–332. MR 850538, DOI 10.1215/S0012-7094-86-05320-2
- V. Bargmann, On a Hilbert space of analytic functions and an associated integral transform, Comm. Pure Appl. Math. 14 (1961), 187–214. MR 157250, DOI 10.1002/cpa.3160140303
- F. F. Bonsall, Hankel operators on the Bergman space for the disc, J. London Math. Soc. (2) 33 (1986), no. 2, 355–364. MR 838646, DOI 10.1112/jlms/s2-33.2.355
- David W. Catlin, Global regularity of the $\bar \partial$-Neumann problem, Complex analysis of several variables (Madison, Wis., 1982) Proc. Sympos. Pure Math., vol. 41, Amer. Math. Soc., Providence, RI, 1984, pp. 39–49. MR 740870, DOI 10.1090/pspum/041/740870
- David W. Catlin and John P. D’Angelo, Positivity conditions for bihomogeneous polynomials, Math. Res. Lett. 4 (1997), no. 4, 555–567. MR 1470426, DOI 10.4310/MRL.1997.v4.n4.a11
- G. B. Folland and J. J. Kohn, The Neumann problem for the Cauchy-Riemann complex, Annals of Mathematics Studies, No. 75, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1972. MR 0461588
- Siqi Fu and Emil J. Straube, Compactness of the $\overline \partial$-Neumann problem on convex domains, J. Funct. Anal. 159 (1998), no. 2, 629–641. MR 1659575, DOI 10.1006/jfan.1998.3317
- S. Fu and E. Straube, Compactness in the $\overline {\partial }$-Neumann problem, Complex Analysis and Geometry (Columbus, OH, 1999), Ohio State Univ. Math. Res. Inst. Publ., 9, de Gruyter, Berlin, pp. 141–160, 2001.
- F. Haslinger, Weighted spaces of entire functions, Indiana Univ. Math. J. 35 (1986), no. 1, 193–208. MR 825636, DOI 10.1512/iumj.1986.35.35011
- Friedrich Haslinger, The canonical solution operator to $\overline \partial$ restricted to Bergman spaces, Proc. Amer. Math. Soc. 129 (2001), no. 11, 3321–3329. MR 1845009, DOI 10.1090/S0002-9939-01-05953-6
- F. Haslinger, The canonical solution operator to $\overline {\partial }$ restricted to spaces of entire functions, Ann. Fac. Sci. Toulouse Math. (6) 11 (2002), 57-70.
- Harro Heuser, Funktionalanalysis, 3rd ed., Mathematische Leitfäden. [Mathematical Textbooks], B. G. Teubner, Stuttgart, 1992 (German). Theorie und Anwendung. [Theory and application]. MR 1219535
- Lars Hörmander, $L^{2}$ estimates and existence theorems for the $\bar \partial$ operator, Acta Math. 113 (1965), 89–152. MR 179443, DOI 10.1007/BF02391775
- Lars Hörmander, An introduction to complex analysis in several variables, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1966. MR 0203075
- Svante Janson, Hankel operators between weighted Bergman spaces, Ark. Mat. 26 (1988), no. 2, 205–219. MR 1050105, DOI 10.1007/BF02386120
- J. J. Kohn and L. Nirenberg, Non-coercive boundary value problems, Comm. Pure Appl. Math. 18 (1965), 443–492. MR 181815, DOI 10.1002/cpa.3160180305
- J. J. Kohn, Harmonic integrals on strongly pseudo-convex manifolds. I, Ann. of Math. (2) 78 (1963), 112–148. MR 153030, DOI 10.2307/1970506
- Steven G. Krantz, Compactness of the $\overline \partial$-Neumann operator, Proc. Amer. Math. Soc. 103 (1988), no. 4, 1136–1138. MR 954995, DOI 10.1090/S0002-9939-1988-0954995-2
- Reinhold Meise and Dietmar Vogt, Einführung in die Funktionalanalysis, Vieweg Studium: Aufbaukurs Mathematik [Vieweg Studies: Mathematics Course], vol. 62, Friedr. Vieweg & Sohn, Braunschweig, 1992 (German). MR 1195130, DOI 10.1007/978-3-322-80310-8
- Richard Rochberg, Trace ideal criteria for Hankel operators and commutators, Indiana Univ. Math. J. 31 (1982), no. 6, 913–925. MR 674875, DOI 10.1512/iumj.1982.31.31062
- Walter Rudin, Functional analysis, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. MR 0365062
- G. Schneider, Compactness of the solution operator to $\overline {\partial }$ on the Fock space in several dimensions, ESI-preprint 1206, 2002.
- Karel Stroethoff, Hankel and Toeplitz operators on the Fock space, Michigan Math. J. 39 (1992), no. 1, 3–16. MR 1137884, DOI 10.1307/mmj/1029004449
- Karel Stroethoff, Compact Hankel operators on the Bergman space, Illinois J. Math. 34 (1990), no. 1, 159–174. MR 1031892
- Karel Stroethoff, Compact Hankel operators on the Bergman spaces of the unit ball and polydisk in $\textbf {C}^n$, J. Operator Theory 23 (1990), no. 1, 153–170. MR 1054822
- Robert Wallstén, Hankel operators between weighted Bergman spaces in the ball, Ark. Mat. 28 (1990), no. 1, 183–192. MR 1049650, DOI 10.1007/BF02387374
- Joachim Weidmann, Lineare Operatoren in Hilberträumen. Teil 1, Mathematische Leitfäden. [Mathematical Textbooks], B. G. Teubner, Stuttgart, 2000 (German). Grundlagen. [Foundations]. MR 1887367, DOI 10.1007/978-3-322-80094-7
- Ke He Zhu, Hilbert-Schmidt Hankel operators on the Bergman space, Proc. Amer. Math. Soc. 109 (1990), no. 3, 721–730. MR 1013987, DOI 10.1090/S0002-9939-1990-1013987-7
Additional Information
- Georg Schneider
- Affiliation: Institut für Mathematik, Universität Wien, Strudlhofgasse 4, A-1090 Wien, Austria
- Address at time of publication: Institut für Betriebswirtschaftslehre, Universität Wien, Brünner Strasse 72, A-1210 Wien, Austria
- Email: georg.schneider@univie.ac.at
- Received by editor(s): October 25, 2002
- Received by editor(s) in revised form: May 15, 2003
- Published electronically: March 24, 2004
- Communicated by: Joseph A. Ball
- © Copyright 2004 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 132 (2004), 2399-2409
- MSC (2000): Primary 47B35; Secondary 32A15
- DOI: https://doi.org/10.1090/S0002-9939-04-07362-9
- MathSciNet review: 2052418