Characterizations of normaloid operators in Hilbert spaces via Birkhoff–James orthogonality

Let $ \mathcal{H} $ be a complex Hilbert space and $ \mathcal{B}(\mathcal{H}) $ the algebra of bounded linear operators on $ \mathcal{H} $. An operator $ T $ is said to be normaloid if its numerical radius $ w(T) $ equals its operator norm $ \|T\| $. In this paper, we establish several characterizations of normaloid operators in Hilbert spaces. In particular, we investigate these operators through the framework of Birkhoff–James orthogonality and norm-parallelism. Mainly, we show that $ T $ is normaloid if, and only if, there exists $ \xi_0 \in \mathbb{C} $ with $ |\xi_0| = \|T\| $ such that$ I \perp_{BJ} (T - \xi_0 I), $where $ \perp_{BJ} $ denotes Birkhoff–James orthogonality. We also present further equivalent formulations and explore various structural consequences of these characterizations.