On a characterization of Riesz bases via biorthogonal sequences (Q2194726)

In this paper it is proved that the completeness of one (any one) of the biorthogonal sequences can be removed from the characterization of a Riesz basis. Concretely, it is shown that \((f_k)_{k=1}^{\infty}\) is a Riesz basis for the Hilbert space \(H\) if and only if \((f_k)_{k=1}^{\infty}\) is a Bessel sequence in \(H\), it has a biorthogonal sequence \((g_k)_{k=1}^{\infty}\) which is a Bessel sequence in \(H\), and one of \((f_k)_{k=1}^{\infty}\) and \((g_k)_{k=1}^{\infty}\) is complete in \(H\). This implies a simpler verification of the Riesz basis property. Moreover, the result is applied to obtain conclusions about certain Gabor systems at the critical density. It is also proved that given two biorthogonal Bessel sequences in \(H\), if one of them is complete in \(H\), then the other one is also complete in \(H\).