Unconditional exponential bases in Hilbert spaces

In the present paper, we consider the existence of unconditional exponential bases in general Hilbert spaces $H=H(E)$ consisting of functions defined on some set $E\subset\mathbb C$ and satisfying the following conditions.
1. The norm in the space $H$ is weaker than the uniform norm on $E$, i.e. the following estimate holds for some constant $A$ and for any function $f$ from $H$:
$$ \|f\|_H\le A\sup_{z\in E}|f(z)|. $$

2. The system of exponential functions $\{\exp(\lambda z),\lambda\in\mathbb C\}$ belongs to the subset $H$ and it is complete in $H$.
It is proved that unconditional exponential bases cannot be constructed in $H$ unless a certain condition is carried out.
Sufficiency of the weakened condition is proved for spaces defined more particularly.