# Spectrum of the Cesaro-Hardy operator in Lorentz Lp,q(0,1) spaces

### Abstract

The aim of this paper is to investigate the spectrum of the Cesaro-Hardy operator in Lorentz Lp,q spaces over (0,1). In this paper we extended Leibowitz's results for Lp space to Lorentz spaces. Note that Lp space is a special case of Lorentz spaces when indexes p and q coincide. Interestingly, we obtained the same results as for Lp space. The point spectrum is obtained by solving an Euler differential equation of first order. We used the operator Pξ to find the resolvent set of the Cesaro-Hardy operator. This operator was defned in Boyd's work in [1]. Boundedness of the operator Pξ on Lp was proved in the same paper. But its boundedness on Lp,q was proved in this paper by using Lp,q norm of the dilation operator. Here, we also used the Boyd's theorem, which describes boundedness of operators on rearrangement invariant spaces. We verified conditions of Boyd's theorem. It allows us to obtain a bounded inverse of the operator λI −C for some complex numbers λ.### References

[1] Boyd, D. W., "Spectrum of Ces ́aro operato" , Acta Scientiarum Mathematicarum, 29.1-2 (1968): 31.

[2] Bennett C., Sharpley R. C., Interpolation of operators (Academic press, 1988).

[3] Brown A., Halmos P. R., Shields A. L., "Ces ́aro operators" , Acta Sci. Math.(Szeged), 26.125-137 (1965): 81-82.

[4] Leibowitz G. M., "Spectra of finite range Ces ́aro operators" , Acta Scientiarum Mathematicarum, 35 (1973): 27-29.

[5] Albanese A. A., Bonet J., Ricker W. J., "On the continuous Ces ́aro operator in certain function spaces" , Positivity, 19.3 (2015): 659-679.

[6] Albanese A. A., "Spectrum of the Cesa ́ro Operator on the Ultradifferentiable Function Spaces Eω(R+)", Complex Analysis and Operator Theory, 15.1 (2021): 1-14.

[7] Dowson H. R., Spectral theory of linear operators No. 12. (Academic Press, 1978).

[8] Kristiansson E., "Decreasing rearrangement and Lorentz L (p, q) spaces [master thesis]", Lulea: Department of

Mathematics of the Lulea University of Technology (2002).

[9] Boyd D. W., "The Hilbert transform on rearrangement-invariant spaces" , Canadian Journal of Mathematics 19 (1967): 599-616.

[10] Boyd, D. W., "Spaces between a pair of reflexive Lebesgue spaces" , Proceedings of the American Mathematical Society 18.2 (1967): 215-219.

[2] Bennett C., Sharpley R. C., Interpolation of operators (Academic press, 1988).

[3] Brown A., Halmos P. R., Shields A. L., "Ces ́aro operators" , Acta Sci. Math.(Szeged), 26.125-137 (1965): 81-82.

[4] Leibowitz G. M., "Spectra of finite range Ces ́aro operators" , Acta Scientiarum Mathematicarum, 35 (1973): 27-29.

[5] Albanese A. A., Bonet J., Ricker W. J., "On the continuous Ces ́aro operator in certain function spaces" , Positivity, 19.3 (2015): 659-679.

[6] Albanese A. A., "Spectrum of the Cesa ́ro Operator on the Ultradifferentiable Function Spaces Eω(R+)", Complex Analysis and Operator Theory, 15.1 (2021): 1-14.

[7] Dowson H. R., Spectral theory of linear operators No. 12. (Academic Press, 1978).

[8] Kristiansson E., "Decreasing rearrangement and Lorentz L (p, q) spaces [master thesis]", Lulea: Department of

Mathematics of the Lulea University of Technology (2002).

[9] Boyd D. W., "The Hilbert transform on rearrangement-invariant spaces" , Canadian Journal of Mathematics 19 (1967): 599-616.

[10] Boyd, D. W., "Spaces between a pair of reflexive Lebesgue spaces" , Proceedings of the American Mathematical Society 18.2 (1967): 215-219.

How to Cite

TULENOV, K.; ZAUR, G..
Spectrum of the Cesaro-Hardy operator in Lorentz Lp,q(0,1) spaces.

**Journal of Mathematics, Mechanics and Computer Science**, [S.l.], v. 117, n. 1, apr. 2023. ISSN 2617-4871. Available at: <https://bm.kaznu.kz/index.php/kaznu/article/view/1216>. Date accessed: 08 june 2023. doi: https://doi.org/10.26577/JMMCS.2023.v117.i1.04.
Section

Mathematics

Keywords
Cesa ́ro-Hardy operator, spectrum, point spectrum, Lorentz Lp,q spaces, rearrangement invariant spaces