On the superadditivity and monotonicity of Gram's inequality and related results (Q1917160)

Let \((H;(\cdot,\cdot))\) be an inner product space over the real or complex number field \(K\) and \(\{x_1,\dots,x_n\}\) a system of vectors in \(H\). Consider the Gram matrix \[ G(x_1,\ldots, x_n) = \left[ \begin{matrix} (x_1, x_1) &\ldots & (x_1, x_n) \\ \vdots & & \vdots \\ (x_n, x_1) &\ldots & (x_n, x_n) \end{matrix} \right] \] and the Gram determinant \(\Gamma(x_1,\dots,x_n) = \text{det} G (x_1, \dots, x_n)\). The following inequality is known as Gram's inequality: \[ \Gamma (x_1, \dots, x_n) \geq 0. \] The main aim of this paper is to point out some new improvements of the classical inequality of Gram as well as to establish a few closely connected results. Applications to linear operators which generalize or improve known results are also given. These inequalities complement in a natural way the results of the book of \textit{D. S. Mitrinović}, \textit{J. E. Pečarić} and \textit{A. M. Fink} [Classical and new inequalities in analysis (1993; Zbl 0771.26009)], and their applications to real or complex numbers complement.