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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 $$r_A(T)=\omega _A(T)=\Vert T\Vert _A$$ for every A-normaloid operator T, where $$\Vert T\Vert _A$$ is denoted to be the A-operator seminorm of T. Moreover, some characterizations of A-normaloid and A-spectraloid operators are given.

Spectral radius of semi-Hilbertian space operators and its applications