Square roots of hyponormal operators

An operator T ε ℒ(H) is called a square root of a hyponormal operator if T² is hyponormal. In this paper, we prove the following results: Let S and T be square roots of hyponormal operators. (1) If σ(T) ∩ [-σ(T)] = φ or {0}, then T is isoloid (i.e., every isolated point of σ(T) is an eigenvalue of T). (2) If S and T commute, then ST is Weyl if and only if S and T are both Weyl. (3) If σ(T) ∩ [-σ(T)] = φ or {0}, then Weyl's theorem holds for T. (4) If σ(T) ∩ [-σ(T)] = φ, then T is subscalar. As a corollary, we get that T has a nontrivial invariant subspace if σ(T) has non-empty interior. (See [3].)