An Explicit Structure for Duals of Frames in Krein Spaces

In this study, motivating the explanation of Esmeral, Ferrer, and Wagner, similar findings regarding frames in Hilbert spaces were attempted to be extended to Krein spaces. The objective of the paper appears to be to explore the properties and characteristics of frames, particularly in the context of Krein spaces and various categories of operators. The paper discusses the implications of removing elements from a frame and the conditions under which a sequence remains a frame or becomes incomplete. In particular, it is shown that the value of $\left[f_{j}, S^{-1}f_{j}\right]$ is important in recognizing which sequences $\{f_{k}\}_{k=1}^{\infty}$ qualify as frames or incomplete sets. Subsequently, the importance of the synthesis operator in characterizing frames and establishing the connection between frame bounds and the properties of the synthesis operator was investigated. Additionally, for certain operators U, conditions have been established under which the sequence $\{Uf_{k}\}_{k=1}^{\infty}$ will be a frame sequence, provided that $\{f_{k}\}_{k=1}^{\infty}$ is already a frame. The concept of a Riesz basis in Krein spaces was introduced, followed by determining the conditions equivalent for a frame to qualify as minimal, a Riesz basis, or an exact frame. Lastly, a clear structure for the duals of frames in Krein spaces has been established. It was discovered that the dual frames of the sequence $\{f_{k}\}_{k=1}^{\infty}$ correspond to the families $\{JV\delta_{k}\}_{k=1}^{\infty}$ where $V:(L^{2}(\mathbb{N}, [.,.]_{\tilde{J}})\rightarrow (K, [.,.]_{J})$ serves as a bounded left inverse of $T_{0}^{*}$ and $J$ is a fundamental symmetry in $K$.