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

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.