Eberlein almost periodic functions that are not pseudo almost periodic

Abstract

We construct Eberlein almost periodic functions $ f_j : J \to H$ so that $||f_1(\cdot)||$ is not ergodic and thus not Eberlein almost periodic and $||f_2(.)||$ is Eberlein almost periodic, but $f_1$ and $f_2$ are not pseudo almost periodic, the Parseval equation for them fails, where $J=\r_+$ or $\r$ and $H$ is a Hilbert space. This answers several questions posed by Zhang and Liu [18].