Square roots of hyponormaloperators

Abstract

An operator $T\in$[Lscr ]$(H)$ 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 $\sigma(T)\cap[-\sigma(T)]=\phi$ 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 $\sigma(T)\cap[-\sigma(T)]=\phi$ or {0}, then Weyl's theorem holds for $T$.

(4) If $\sigma(T)\cap[-\sigma(T)]=\phi$, 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].)