Spectrum of the Hilbert transform on Orlicz spaces over R
DOI:
https://doi.org/10.26577/JMMCS202412111Keywords:
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.
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