| Absztrakt: | Let ${\cal H}$ be a separable Hilbert space. Suppose that $A\in
{\cal L}({\cal H})$ and $A^n$ is normal for some $n\geq 2$. The
author determines consequences for $A$ for each of the following
conditions: \par
1. $A$ is of class ${\cal N}+{\cal C}$, where
${\cal N}$ is the set of normal operators and ${\cal C}$ is the
set of compact operators; \par
2. $A$ is of class ${\cal N}+{\cal
C}$ and $A^n$ is spectral continuous, i.e., $A^n$ has empty point
spectrum; \par
3. $A^n= \lambda I$; \par
4. $A^n=0$; \par
5.
$A$ is a power bounded operator, i.e., $\sup\{|A^j|: j\in\bbfN\}$
is finite. \par
For example, it is shown that in Case 2 it follows
that $A$ must itself be a spectral continuous normal operator.
This result implies that essentially normal roots of spectral
continuous normal operators are normal. ($A$ is essentially normal
if $A^*A-AA^*\in{\cal C}$.) \par
The present note is an extension
of results of {\it B. P. Duggal} [Bull. Lond. Math. Soc. 25,
No. 1, 74-80 (1993; Zbl 0789.47020)].}
\RV{R.C.Gilbert (Placentia) |