Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

ASCENT AND DESCENT OF COMPOSITION OPERATORS ON LORENTZ SPACES

  • Received : 2021.01.18
  • Accepted : 2021.07.23
  • Published : 2022.01.31
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we provide various characterizations for the composition operator on Lorentz spaces L(p, q), 1 < p ≤ ∞, 1 ≤ q ≤ ∞ to have finite ascent (descent) in terms of its inducing measurable transformation. At the end, in order to demonstrate our outcomes, some examples are given.

Keywords

Acknowledgement

The suggestions and comments of the referee(s) to shape the paper in the present form on revision, are gratefully acknowledged.

References

  1. S. C. Arora, G. Datt, and S. Verma, Composition operators on Lorentz spaces, Bull. Austral. Math. Soc. 76 (2007), no. 2, 205-214. https://doi.org/10.1017/S0004972700039599
  2. S. C. Arora, G. Datt, and S. Verma, Multiplication and composition operators on Lorentz-Bochner spaces, Osaka J. Math. 45 (2008), no. 3, 629-641. http://projecteuclid.org/euclid.ojm/1221656645
  3. S. C. Arora, G. Datt, and S. Verma, Operators on Lorentz sequences spaces, Math. Bohem. 134 (2009), no. 1, 87-98. https://doi.org/10.21136/mb.2009.140643
  4. J. T. Campbell and J. E. Jamison, The analysis of composition operators on Lp and the Hopf decomposition, J. Math. Anal. Appl. 159 (1991), no. 2, 520-531. https://doi.org/10.1016/0022-247X(91)90212-I
  5. H. Chandra and P. Kumar, Ascent and descent of composition operators on lp spaces, Demonstratio Math. 43 (2010), no. 1, 161-165. https://doi.org/10.1515/dema-2010-0116
  6. Y. Estaremi, S. Maghsoudi, and I. Rahmani, A note on multiplication and composition operators in Orlicz spaces, Filomat 32 (2018), no. 7, 2693-2699. https://doi.org/10.2298/fil1807693e
  7. R. K. Giri and S. Pradhan, On the properties of ascent and descent of composition operator on Orlicz spaces, Math. Sci. Appl. E-Notes 5 (2017), no. 1, 70-76.
  8. R. A. Hunt, On L(p, q) spaces, Enseign. Math. (2) 12 (1966), 249-276.
  9. G. G. Lorentz, Some new functional spaces, Ann. of Math. (2) 51 (1950), 37-55. https://doi.org/10.2307/1969496
  10. G. G. Lorentz, On the theory of spaces λ, Pacific J. Math. 1 (1951), 411-429. http://projecteuclid.org/euclid.pjm/1103052109 https://doi.org/10.2140/pjm.1951.1.411
  11. F. Riesz, Uber lineare Funktionalgleichungen, Acta Math. 41 (1916), no. 1, 71-98. https://doi.org/10.1007/BF02422940
  12. R. K. Singh and J. S. Manhas, Composition operators on function spaces, North-Holland Mathematics Studies, 179, North-Holland Publishing Co., Amsterdam, 1993.
  13. A. E. Taylor, Introduction to Functional Analysis, John Wiley & Sons, Inc., New York, 1958.
  14. A. E. Taylor, Theorems on ascent, descent, nullity and defect of linear operators, Math. Ann. 163 (1966), 18-49. https://doi.org/10.1007/BF02052483