Spectrum of the Hilbert transform on Orlicz spaces over R

Authors

DOI:

https://doi.org/10.26577/JMMCS202412111
        220 181

Keywords:

Hilbert transfrom, spectrum, point spectrum, Orlicz space.

Abstract

In this paper, we investigate the spectrum of the classical Hilbert transform on Orlicz spaces LΦ over the real line R, extending Widom's and Jorgens's results in the context of Lp spaces [3, 8], since the classical Lebesgue spaces are particular examples of Orlicz spaces when the N-function
Φ = xp/p. Our motivation to do so is due to the classical result of Boyd [1] which says that the Hilbert transform is bounded on certain Orlicz spaces and the fact that the spectrum of the bounded linear operator is not an empty set. We rst present an auxiliary result from the general
theory of Banach algebras and results from general theory of Banach spaces, which further helps us to give a full decsription of the ne spectrum of the Hilbert transform on Orlicz spaces over the real line R. We also present a resolvent set of the Hilbert transform on Orlicz spaces over the real line R as well as its resolvent operator.

References

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

Bennett C., Sharpley R. C., "Interpolation of operators."Academic press, (1988).

Jorgens K., "Linear integral operators" , Boston: Pitman Advanced Publishing Program, 26.125-137 (1982).

Krasnoselskii M.A., Rutitskii Ya.B., "Convex Functions and Orlicz Spaces" , Fizmatgiz, Moscow(1958).

Rao M.M., Ren Z.D., "Theory of Orlicz spaces" , Chapman & Hall Pure and Applied Mathematics, (1991).

Boyd D. W., "The Hilbert Transformation on rearrangement invariant Banach spaces" , Thesis, University of Toronto, (1966).

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

Widom H., "Singular integral equations in Lp" , Trans. Am. Math. Soc. 97 (1960): 131-160.

Curbera G., Okado S. Ricker W., "Fine spectra of the nite Hilbert transform in function spaces" , Advances in Mathematics 380 (2021). 10. Fernandes C. A., Karlovich A. Yu., Karlovich Yu. I., "Calkin Images of Fourier Convolution Operators with Slowly Oscillating Symbols" , Operator Theory, Functional Analysis and Applications 282 (2021): 193-218.

King F.W. "Hilbert Transforms, vol.I," , Cambridge: Cambridge University Press 898 (2009)

Downloads

How to Cite

Akhymbek, M., Tastankul, . R., & Ozbekbay, B. (2024). Spectrum of the Hilbert transform on Orlicz spaces over R. Journal of Mathematics, Mechanics and Computer Science, 121(1), 3–11. https://doi.org/10.26577/JMMCS202412111