Some inequalities in Pseudo-Hilbert spaces

Abstract

The aim of this paper is to obtain new versions of the reverse of the generalized triangle inequalities given in \cite{SSDNA}, %[4],and \cite{SSDPR} %[5] if the pair $(a_i,x_i),\;i\in\{1,\ldots,n\}$ from Theorem 1 of \cite{SSDNA} %[4] belongs to ${\mathbb C}\times\mathcal H $, where $\mathcal H$ is a Loynes $Z$-space instead of ${\mathbb K}\times X$, $X$ being a normed linear space and ${\mathbb K}$ is the field of scalars. By comparison, in \cite{SSDNA} %[4] the pair $(a_i,x_i),\;i\in\{1,\ldots,n\}$ belongs to $A^2$, where $A$ is a normed algebra over the real or complex number field ${\mathbb K}.$ The results will be given in Theorem 1, Theorem 3, Remark 2 and Corollary 3 which represent other interesting variants of Theorem 2.1, Remark 2.2, Theorem 3.2 and Theorem 3.4., see \cite{SSDNA}. %[4].