$A$-spectral permanence property for $C^{\ast }$-algebras

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