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 $\Vert T\Vert$. 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 $|{\xi }_0|=\Vert T\Vert$ 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.