Riesz basis property of evolution equations in Hilbert spaces and application to a coupled string equation

Suppose that {lambda(n)} is the set of zeros of a sine-type generating function of the exponential system {e(ilambdant)} in L-2(0, T) and is separated. Levin and Golovin's classical theorem claims that {e(ilambdant)} forms a Riesz basis for L-2(0, T). In this article, we relate this result with Riesz basis generation of eigenvectors of the system operator of the linear time-invariant evolution equation in Hilbert spaces through its spectrum. A practically favorable necessary and sufficient condition for the separability of zeros of function of sine type is derived. The result is applied to get Riesz basis generation of a coupled string equation with joint dissipative feedback control.