  In this paper, we prove the relation $\frac{r_{A}(T) + r_{A}(T^{\diamond}) +
|r_{A}(T^{\diamond}) - r_{A}(T)|}{2} = \sup \{ |\lambda|: \lambda \in
\sigma_{A}(T)\}$, where $A$ is a positive semidefinite operator (not
necessarily to have a closed range) and $r_{A}(T)$ is the $A$-spectral radius
of $T$ in $B_{A^{\frac{1}{2}}}(H)$. Also we prove that $\sup \{ |\lambda|:
\lambda \in \sigma_{A}(T)\} = r_{A}(T), \text{ when } T \in
B_{A^{\frac{1}{2}}}(H) \text { commutes with } A$. By introducing $A$-Harte
spectrum $\sigma_{A_{h}}(\mathbf{T})$ of a $d$-tuple operator $\mathbf{T}=
(T_{1},\dots,T_{d}) \in (B_{A^{\frac{1}{2}}}(H))^{d}$, we prove that
$r_{A_{h}}(\mathbf{T}) \leq \sup \{\|\lambda\|_{2}: \lambda \in
\sigma_{A_{h}}(\mathbf{T})\}$, where $r_{A_{h}}(\mathbf{T})$ is the $A$-Harte
spectral radius of $\mathbf{T}$.
