Square roots of hyponormal operators

Abstract

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