Spectral radius of semi-Hilbertian space operators and its applications

In this paper, we aim to introduce the notion of the spectral radius of bounded linear operators acting on a complex Hilbert space $\mathcal{H}$, which are bounded with respect to the seminorm induced by a positive operator $A$ on $\mathcal{H}$. Mainly, we show that $r_A(T)\le {\omega }_A(T)$ for every $A$-bounded operator $T$, where $r_A(T)$ and ${\omega }_A(T)$ denote respectively the $A$-spectral radius and the $A$-numerical radius of $T$. This allows to establish that $r_A(T)={\omega }_A(T)=\mathrm{\parallel }T{\mathrm{\parallel }}_A$ for every $A$-normaloid operator $T$, where $\mathrm{\parallel }T{\mathrm{\parallel }}_A$ is denoted to be the $A$-operator seminorm of $T$. Moreover, some characterizations of $A$-normaloid and $A$-spectraloid operators are given.