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

Abstract

Let H be a complex Hilbert space and B(H) the algebra of bounded linear operators on H. An operator T is said to be normaloid if its numerical radius w(T) equals its operator norm kTk. 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 normparallelism. Mainly, we show that T is normaloid if, and only if, there exists ξ0 ∈ C with |ξ0| = kTk such that I ⊥BJ (T − ξ0I), where ⊥BJ denotes Birkhoff–James orthogonality. We also present further equivalent formulations and explore various structural consequences of these characterizations.