Let \(\mathcal{B(H)}\) be the algebra of all bounded operators defined on a complex Hilbert space \(\mathcal{H}\). An operator \(S \in \mathcal{B(H)}\) is said to be normaloid if \(r(S)=\|S\|\) where \(r(S)\) denotes the spectral radius of \(S\). For an operator \(S \in \mathcal{B(H)}\), there exists a unique polar decomposition \(S=U|S|\), where \(|S|=(S^{\ast}S)^{\frac{1}{2}}\) and \(U\) is the partial isometry satisfying \(\ker U=\ker S\). Under this polar decomposition, the \(\lambda\)-Aluthge transform of \(S\) is defined by \(\Delta_{\lambda}S=|S|^{\lambda}U|S|^{1-\lambda}\), \(\lambda \in [0,1]\). In particular, when \(\lambda=\frac{1}{2}\), it is called the Aluthge transform of \(S\), denoted \(\Delta_{1/2}S=|S|^{\frac{1}{2}}U|S|^{\frac{1}{2}}\). Moreover, the \(n\)th iterated Aluthge transform of \(S\) is defined as \(\Delta_{1/2}^{n}(S)=\Delta(\Delta_{1/2}^{n-1}(S))\) and \(\Delta_{1/2}^{1}(S)=\Delta(S)\) for any \(n \in \mathbb{N}\). Furthermore, the generalized Aluthge transform of \(S\) is defined as \(\Delta_{f,g}S=f(|S|)Ug(|S|)\), where \(f\) and \(g\) are continuous functions such that \(g(x)f(x)=x\), \(x \geq 0\).
In this paper, the authors express the spectral radius of \(S\) in \(\mathcal{B(H)}\) by using the properties of the generalized Aluthge transform of \(S\), the asymptotic behavior of powers of \(S\), the numerical radius, and the iterated generalized Aluthge transform. As some applications, they apply their results to normaloid operators.