The parabolic algebra on Lp spaces

Abstract

The parabolic algebra was introduced by Katavolos and Power, in 1997, as the SOT-closed operator algebra acting on L2(R) that is 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 in the Volterra nest and its analytic counterpart. We prove that a similar result holds for the corresponding algebras acting on Lp(R), where 1<p<∞. In the last section, it is also shown that the reflexive closures of the Fourier binests on Lp(R) are all order isomorphic for 1<p<∞.