One remark about the unconditional exponential bases and cosine bases, connected with them - Volkova, M. G.; Gubreev, G. M. (Methods of Functional Analysis and Topology, 2008)
In the paper we consider examples of basis families {λ_k t}^∞_1, λ_k>0, in the space L_2(0,σ), such that systems {e^iλ_kt,e^-iλ_kt}^∞_1 don’t form an unconditional basis in space L_2(-σ,σ).