Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

ESSENTIAL NORMS OF INTEGRAL OPERATORS

  • Received : 2018.04.14
  • Accepted : 2018.10.11
  • Published : 2019.03.01
Download PDF
⟨ Previous Next ⟩

Abstract

We estimate the essential norms of Volterra-type integral operators $V_g$ and $I_g$, and multiplication operators $M_g$ with holomorphic symbols g on a large class of generalized Fock spaces on the complex plane ${\mathbb{C}}$. The weights defining these spaces are radial and subjected to a mild smoothness conditions. In addition, we assume that the weights decay faster than the classical Gaussian weight. Our main result estimates the essential norms of $V_g$ in terms of an asymptotic upper bound of a quantity involving the inducing symbol g and the weight function, while the essential norms of $M_g$ and $I_g$ are shown to be comparable to their operator norms. As a means to prove our main results, we first characterized the compact composition operators acting on the spaces which is interest of its own.

Keywords

References

  1. A. Aleman, A class of integral operators on spaces of analytic functions, in Topics in complex analysis and operator theory, 3-30, Univ. Malaga, Malaga, 2007.
  2. B. J. Carswell, B. D. MacCluer, and A. Schuster, Composition operators on the Fock space, Acta Sci. Math. (Szeged) 69 (2003), no. 3-4, 871-887.
  3. S. Chandrasekhar, Radiative Transfer, Oxford University Press, 1950.
  4. O. Constantin, A Volterra-type integration operator on Fock spaces, Proc. Amer. Math. Soc. 140 (2012), no. 12, 4247-4257. https://doi.org/10.1090/S0002-9939-2012-11541-2
  5. O. Constantin and J. Pelaez, Integral operators, embedding theorems and a Littlewood- Paley formula on weighted Fock spaces, J. Geom. Anal. 26 (2016), no. 2, 1109-1154. https://doi.org/10.1007/s12220-015-9585-7
  6. C. Corduneanu, Integral Equations and Applications, Cambridge University Press, Cambridge, 1991.
  7. Z. Cuckovic and R. Zhao, Weighted composition operators on the Bergman space, J. London Math. Soc. (2) 70 (2004), no. 2, 499-511. https://doi.org/10.1112/S0024610704005605
  8. Z. Cuckovic and R. Zhao, Weighted composition operators between different weighted Bergman spaces and different Hardy spaces, Illinois J. Math. 51 (2007), no. 2, 479-498. https://doi.org/10.1215/ijm/1258138425
  9. K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, 1985.
  10. S. Hu, M. Khavanin, and W. Zhuang, Integral equations arising in the kinetic theory of gases, Appl. Anal. 34 (1989), no. 3-4, 261-266. https://doi.org/10.1080/00036818908839899
  11. S. Janson, J. Peetre, and R. Rochberg, Hankel forms and the Fock space, Rev. Mat. Iberoamericana 3 (1987), no. 1, 61-138.
  12. T. Mengestie, Volterra type and weighted composition operators on weighted Fock spaces, Integral Equations Operator Theory 76 (2013), no. 1, 81-94. https://doi.org/10.1007/s00020-013-2050-8
  13. T. Mengestie, Product of Volterra type integral and composition operators on weighted Fock spaces, J. Geom. Anal. 24 (2014), no. 2, 740-755. https://doi.org/10.1007/s12220-012-9353-x
  14. T. Mengestie, Carleson type measures for Fock-Sobolev spaces, Complex Anal. Oper. Theory 8 (2014), no. 6, 1225-1256. https://doi.org/10.1007/s11785-013-0321-7
  15. T. Mengestie and S. I. Ueki, Integral, differential and multiplication operators on weighted Fock spaces, Integral, differential and multiplication operators on weighted Fock spaces, Complex Anal. Oper. Theory. DOI: 10.1007/s11785-018-0820-7.
  16. D. O'Regan and M. Meehan, Existence Theory for Nonlinear Integral and Integrodiffer- ential Equations, Mathematics and its Applications, 445, Kluwer Academic Publishers, Dordrecht, 1998.
  17. J. Pau and J. Pelaez, Embedding theorems and integration operators on Bergman spaces with rapidly decreasing weights, J. Funct. Anal. 259 (2010), no. 10, 2727-2756. https://doi.org/10.1016/j.jfa.2010.06.019
  18. J. Rattya, The essential norm of a composition operator mapping into the Qs-space, J. Math. Anal. Appl. 333 (2007), no. 2, 787-797. https://doi.org/10.1016/j.jmaa.2006.11.010
  19. J. H. Shapiro, The essential norm of a composition operator, Ann. of Math. (2) 125 (1987), no. 2, 375-404. https://doi.org/10.2307/1971314
  20. A. G. Siskakis, Volterra operators on spaces of analytic functions a survey, in Proceedings of the First Advanced Course in Operator Theory and Complex Analysis, 51-68, Univ. Sevilla Secr. Publ., Seville, 2006.
  21. S. Stevic, Weighted composition operators between Fock-type spaces in CN, Appl. Math. Comput. 215 (2009), no. 7, 2750-2760. https://doi.org/10.1016/j.amc.2009.09.016
  22. S.-I. Ueki, Weighted composition operators on some function spaces of entire functions, Bull. Belg. Math. Soc. Simon Stevin 17 (2010), no. 2, 343-353. https://doi.org/10.36045/bbms/1274896210