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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Spectral properties of linear operators for which $T^*T$ and $T$ $+$ $T^*$ commute


Authors: Stephen L. Campbell and Ralph Gellar
Journal: Proc. Amer. Math. Soc. 60 (1976), 197-202
MSC: Primary 47B20
DOI: https://doi.org/10.1090/S0002-9939-1976-0417841-3
MathSciNet review: 0417841
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Abstract: The class of linear operators for which ${T^\ast }T$ and $T + {T^\ast }$ commute is studied. It is shown that such operators are normaloid. If T is also completely nonnormal, then $\sigma (T) = \sigma ({T^\ast })$. Also, isolated points of $\sigma (T)$ are reducing eigenvalues. Finally, if $\sigma (T)$ is a subset of either a vertical line or the real axis, then T is normal.


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Keywords: Operator such that <IMG WIDTH="44" HEIGHT="21" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="${T^\ast }T$"> and <!– MATH $T + {T^\ast }$ –> <IMG WIDTH="68" HEIGHT="38" ALIGN="MIDDLE" BORDER="0" SRC="images/img2.gif" ALT="$T + {T^\ast }$"> commute, spectrum, normaloid operator, spectraloid operator, isoloid operator
Article copyright: © Copyright 1976 American Mathematical Society