Kérchy László honlapja

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Cím:On roots of normal operators.
Szerző:K\'erchy, L\'aszl\'o
Forrás:Acta Sci. Math. 60, No.3-4, 439-449 (1995).
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) 
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