Geodesic
Unconditional exponential bases in Hilbert spaces
Ufa mathematical journal, Tome 3 (2011) no. 1, pp. 3-15

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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.
Keywords: series of exponents, unconditional bases, Hilbert space. 
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     author = {K. P. Isaev and R. S. Yulmukhametov},
     title = {Unconditional exponential bases in {Hilbert} spaces},
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K. P. Isaev; R. S. Yulmukhametov. Unconditional exponential bases in Hilbert spaces. Ufa mathematical journal, Tome 3 (2011) no. 1, pp. 3-15. http://geodesic.mathdoc.fr/item/UFA_2011_3_1_a0/