The parabolic algebra on Banach spaces

TL;DR

This paper extends the study of the parabolic algebra from Hilbert spaces to Banach spaces, specifically on L^p(R), showing reflexivity and isomorphism properties similar to the original setting.

Contribution

It proves that the reflexivity and algebraic properties of the parabolic algebra on L^2(R) also hold on L^p(R) for 1<p<∞, generalizing previous results.

Findings

The parabolic algebra on L^p(R) is reflexive.

The Fourier binest algebras on L^p(R) are order isomorphic.

Similar algebraic structures hold across different p-values.

Abstract

The parabolic algebra was introduced by Katavolos and Power, in 1997, as the operator algebra acting on $L^2(R)$ that is weakly generated by the translation and multiplication semigroups. In particular, they proved that this algebra is reflexive and is equal to the Fourier binest algebra, that is, to the algebra of operators that leave invariant the subspaces of the Volterra nest and its analytic counterpart. We prove that a similar result holds for the corresponding algebras acting on $L^p(R)$, where $1<p<\infty$. It is also shown that the reflexive closures of the Fourier binests on $L^p(R)$ are all order isomorphic for $1 < p < \infty$.