Frames and Dual Frames for Krein Spaces (Q6593315)

Frames for Hilbert spaces are a generalization of bases. They have several applications in mathematics and engineering. Krein spaces generalize Hilbert spaces and are closely related to quantum field theory.\N\NIn 2002, \textit{I. Peng} and \textit{S. Waldron} [Linear Algebra Appl. 347, No. 1--3, 131--157 (2002; Zbl 0999.05014)] defined signed frames as a generalization of tight frames. Based on this, in 2012, \textit{J. I. Giribet} et al. [J. Math. Anal. Appl. 393, No. 1, 122--137 (2012; Zbl 1253.46033)] defined J-frames for Krein spaces. In 2015, \textit{K. Esmeral} et al. [Banach J. Math. Anal. 9, No. 1, 1--16 (2015; Zbl 1311.42076)] proposed another concept of frames for Krein spaces which covers that of J-frames. This new concept is more directly dependent on the structure of Krein spaces and avoids switching between Krein spaces and Hilbert spaces.\N\NThe present paper continues the study of frames in Krein spaces \(\mathcal K\) proposed by Esmeral et al. [loc. cit.]. The authors establish a link among J-orthonormal bases, Parseval frames, and orthonormal bases for \(\mathcal K\). They present the reconstruction formula associated with a Parseval frame for \(\mathcal K\). This reconstruction formula shows that a Parseval frame for genuine Krein spaces cannot be a dual frame of its own. They give examples to illustrate that a J-orthonormal basis (resp. Parseval frame) is not necessarily a Parseval frame (resp. J-orthonormal basis) and that a Parseval frame and a J-orthonormal basis need not admit the same expansion for vectors in \(\mathcal K\). They prove that the canonical dual of a frame for K gives an expression of vectors in \(\mathcal K\) with coefficients having minimal 2-norm. They show that removing one vector from a frame for \(\mathcal K\) leaves either a frame or an incomplete set. They prove that all frames for \(\mathcal K\) can be obtained by applying a bounded and surjective operator on \(\mathcal K\) to a J-orthonormal basis. They present a parametric expression of all dual frames of a given frame for \(\mathcal K\).