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].)