Riesz Bases in Krein Spaces

Shah Jahan () and P. Sam Johnson ()
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Shah Jahan: Central University of Haryana
P. Sam Johnson: National Institute of Technology Karnataka (NITK)

Indian Journal of Pure and Applied Mathematics, 2025, vol. 56, issue 3, 1005-1013

Abstract: Abstract We start by introducing and studying the definition of a Riesz basis in a Krein space $$({\mathcal {K}},[.,.])$$ ( 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 $$\{[f_n,f_j]_{n,j \in I_+}\}$$ { [ f n , f j ] n , j ∈ I + } and $$\{[f_n,f_j]_{n,j \in I_-}\}$$ { [ f n , f j ] n , j ∈ I - } become bounded invertible operators. Finally, we provide an equivalent characterization of a Riesz basis in terms of Gram matrices.

Keywords: Krein space; Riesz basis; biorthogonal sequence; Gram matrix; frame sequence; 42C15; 46C05; 46C20 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s13226-025-00816-3

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