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On results of Krein, Rao and Lin about angles between vectors in a Hilbert space

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Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

Abstract

In this paper the problem of refining Krein-Rao (K-R) and related inequalities is discussed. A collection of identities induced by K-R functional and corresponding K-R inequalities of second kind are derived. Such results are applied to improve the original K-R inequality. A new lower estimate of the functional is deduced. An interpretation of the obtained refinement is shown as the monotonicity of the K-R functional. An application for numerical radius of bounded linear operators is also given. A method due to Sababheh et al. (Oper Matrices 16(1):16–19 2022) is generalized from the cosine function to a wider class of functions attending some special decompositions. In consequence, analogous results are shown for the power function and hyperbolic cosinus in place of cosine. In addition, a refinement of a Dragomir’s (Linear Multilinear Algebra 67(2) 337–347 2019) improvement of Schwarz inequality is provided.

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Acknowledgements

The author would like to thank anonymous referees for giving valuable comments that helped to improve the previous version of the paper.

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  1. Institute of Mathematics, Pedagogical University of Cracow, Podchora̧żych 2, 30-084, Kraków, Poland

    Marek Niezgoda

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  1. Marek Niezgoda
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Niezgoda, M. On results of Krein, Rao and Lin about angles between vectors in a Hilbert space. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 117, 73 (2023). https://doi.org/10.1007/s13398-023-01405-x

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