A Volterra-type integration operator on Fock spaces

Constantin, Olivia

doi:10.1090/S0002-9939-2012-11541-2

A Volterra-type integration operator on Fock spaces
HTML articles powered by AMS MathViewer

by Olivia Constantin PDF

Proc. Amer. Math. Soc. 140 (2012), 4247-4257 Request permission

Abstract:

We study certain spectral properties and the invariant subspaces for some classes of integration operators of Volterra type on the Fock space.

References

Alexandru Aleman, A class of integral operators on spaces of analytic functions, Topics in complex analysis and operator theory, Univ. Málaga, Málaga, 2007, pp. 3–30. MR 2394654
Alexandru Aleman and Joseph A. Cima, An integral operator on $H^p$ and Hardy’s inequality, J. Anal. Math. 85 (2001), 157–176. MR 1869606, DOI 10.1007/BF02788078
Alexandru Aleman and Olivia Constantin, Spectra of integration operators on weighted Bergman spaces, J. Anal. Math. 109 (2009), 199–231. MR 2585394, DOI 10.1007/s11854-009-0031-2
Alexandru Aleman and Boris Korenblum, Volterra invariant subspaces of $H^p$, Bull. Sci. Math. 132 (2008), no. 6, 510–528. MR 2445578, DOI 10.1016/j.bulsci.2007.08.001
Alexandru Aleman and Aristomenis G. Siskakis, Integration operators on Bergman spaces, Indiana Univ. Math. J. 46 (1997), no. 2, 337–356. MR 1481594, DOI 10.1512/iumj.1997.46.1373
Alexandru Aleman and Aristomenis G. Siskakis, An integral operator on $H^p$, Complex Variables Theory Appl. 28 (1995), no. 2, 149–158. MR 1700079, DOI 10.1080/17476939508814844
H.-R. Cho and K. Zhu, Fock-Sobolev spaces and their Carleson measures, preprint available at http://www.albany.edu/$\sim$kzhu/cho-zhu2.pdf.
Olivia Constantin, Carleson embeddings and some classes of operators on weighted Bergman spaces, J. Math. Anal. Appl. 365 (2010), no. 2, 668–682. MR 2587070, DOI 10.1016/j.jmaa.2009.11.035
Petros Galanopoulos, Daniel Girela, and José Ángel Peláez, Multipliers and integration operators on Dirichlet spaces, Trans. Amer. Math. Soc. 363 (2011), no. 4, 1855–1886. MR 2746668, DOI 10.1090/S0002-9947-2010-05137-2
Daniel Girela and José Ángel Peláez, Carleson measures, multipliers and integration operators for spaces of Dirichlet type, J. Funct. Anal. 241 (2006), no. 1, 334–358. MR 2264253, DOI 10.1016/j.jfa.2006.04.025
I. C. Gohberg and M. G. Kreĭn, Introduction to the theory of linear nonselfadjoint operators, Translations of Mathematical Monographs, Vol. 18, American Mathematical Society, Providence, R.I., 1969. Translated from the Russian by A. Feinstein. MR 0246142
Z. J. Hu, Equivalent norms on Fock spaces with some application to extended Cesaro operators, preprint (May 2011). To appear in Proc. Amer. Math. Soc.
Svante Janson, Jaak Peetre, and Richard Rochberg, Hankel forms and the Fock space, Rev. Mat. Iberoamericana 3 (1987), no. 1, 61–138. MR 1008445, DOI 10.4171/RMI/46
Daniel H. Luecking, Embedding theorems for spaces of analytic functions via Khinchine’s inequality, Michigan Math. J. 40 (1993), no. 2, 333–358. MR 1226835, DOI 10.1307/mmj/1029004756
N. K. Nikol′skiĭ, Invariant subspaces of weighted shift operators, Mat. Sb. (N.S.) 74 (116) (1967), 172–190 (Russian). MR 0229081
N. K. Nikol′skiĭ, Unicellularity and non-unicellularity of weighted shift operators, Dokl. Akad. Nauk SSSR 172 (1967), 287–290 (Russian). MR 0217639
Jordi Pau and José Ángel Peláez, Embedding theorems and integration operators on Bergman spaces with rapidly decreasing weights, J. Funct. Anal. 259 (2010), no. 10, 2727–2756. MR 2679024, DOI 10.1016/j.jfa.2010.06.019
Ch. Pommerenke, Schlichte Funktionen und analytische Funktionen von beschränkter mittlerer Oszillation, Comment. Math. Helv. 52 (1977), no. 4, 591–602 (German). MR 454017, DOI 10.1007/BF02567392
Jouni Rättyä, Integration operator acting on Hardy and weighted Bergman spaces, Bull. Austral. Math. Soc. 75 (2007), no. 3, 431–446. MR 2331020, DOI 10.1017/S0004972700039356
Aristomenis G. Siskakis, Volterra operators on spaces of analytic functions—a survey, Proceedings of the First Advanced Course in Operator Theory and Complex Analysis, Univ. Sevilla Secr. Publ., Seville, 2006, pp. 51–68. MR 2290748
Robert Wallstén, The $S^p$-criterion for Hankel forms on the Fock space, $0<p<1$, Math. Scand. 64 (1989), no. 1, 123–132. MR 1036432, DOI 10.7146/math.scand.a-12251