On some norm equalities in pre-Hilbert \(C^{\ast}\)-modules
Hilbert \(C^{*}\)-modules are generalizations of Hilbert spaces by allowing the inner product to take values in a \(C^{*}\)-algebra rather than in the field of complex numbers, but there are some differences between these two classes. For example, a closed submodule of a Hilbert \(C^{*}\)-module is not necessary complemented. In this paper, the authors investigate some norm equalities in Hilbert \(C^{*}\) -modules. The triangle inequality \(\left\| x+y\right\| \leq \left\| x\right\| +\left\| y\right\| \) in normed linear spaces is well-known, as is the fact that for two elements \(x\) and \(y\) in a Hilbert space, \(\left\| x+y\right\| =\left\| x\right\| +\left\| y\right\| \) if and only if \(\left\langle x,y\right\rangle =\left\| x\right\| \left\| y\right\| .\) The authors extend this result in the context of Hilbert \(C^{*}\)-modules and show that, given a Hilbert \(C^{*}\)-module \(X\) and two elements \(x\) and \(y\) in \(X\), \(\left\| x+y\right\| =\left\| x\right\| +\left\| y\right\| \) if and only if \(\left\| x\right\| \left\| y\right\| \) is an element in the numerical range of \(\left\langle x,y\right\rangle \). Using this result, the authors obtain a characterization of Pythagoras' equality in a Hilbert \(C^{*}\)-module.
scientific article; zbMATH DE number 5016218