Riesz Bases in Krein Spaces

Abstract: We start by introducing and studying the definition of a Riesz basis in a Krein space $(\mathcal{K},[.,.])$, along with a condition under which a Riesz basis becomes a Bessel sequence. The concept of biorthogonal sequence in Krein spaces is also introduced, providing an equivalent characterization of a Riesz basis. Additionally, we explore the concept of the Gram matrix, defined as the sum of a positive and a negative Gram matrices, and specify conditions under which the Gram matrix becomes bounded in Krein spaces. Further, we characterize the conditions under which the Gram matrices ${[f_n,f_j]{n,j \in I+}}$ and ${[f_n,f_j]{n,j \in I-}}$ become bounded invertible operators. Finally, we provide an equivalent characterization of a Riesz basis in terms of Gram matrices.