A-Spectral Permanence Property for \(C^*\)-Algebras

For a positive element A of a \(C^*\) $C^{\ast }$ -algebra \(\mathfrak {A}\) $\mathfrak{A}$ , let \({\Vert X\Vert }_{A}\) ${\Vert X\Vert }_A$ denote the A-operator semi-norm of \(X\in \mathfrak {A}\) $X\in \mathfrak{A}$ . In this paper, we aim to introduce and study the notion of A-spectrum for X, such that \({\Vert X\Vert }_{A}<\infty \) ${\Vert X\Vert }_A<\infty$ . In particular, when A is well supported, we establish an A-spectral permanence property for \(C^*\) $C^{\ast }$ -algebras.