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 of 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 : parallel to f parallel to(H) <= A sup(z is an element of E) vertical bar f (z) vertical bar. 2. The system of exponential functions {exp(lambda z), lambda is an element of 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.