Square roots of hyponormaloperators

Kim, Mee-Kyoung ; Ko, Eungil

Glasgow mathematical journal, Tome 41 (1999) no. 3, pp. 463-470

Résumé

An operator$T\in$[Lscr]$(H)$ is called a square root of a hyponormaloperator if $T^2$ is hyponormal. In this paper, we prove the followingresults: Let $S$ and $T$ be square roots of hyponormaloperators.(1) If $\sigma(T)\cap[-\sigma(T)]=\phi$ or {0}, then$T$ is isoloid (i.e., every isolated point of $\sigma(T)$ is aneigenvalue of $T$).(2) If $S$ and $T$ commute, then $ST$ is Weylif 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 $T$ issubscalar. As a corollary, we get that $T$ has a nontrivial invariantsubspace if $\sigma(T)$ has non-empty interior. (See[3].)

Détail

DOI : 10.1017/S0017089599000178