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Proceedings of the American Mathematical Society

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

 
 

 

An inequality concerning the smallest disc that contains the spectrum of an operator


Author: Gerhard Garske
Journal: Proc. Amer. Math. Soc. 78 (1980), 529-532
MSC: Primary 47A10
DOI: https://doi.org/10.1090/S0002-9939-1980-0556626-5
MathSciNet review: 556626
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Abstract: Any bounded linear operator T on a Hilbert space satisfies the inequality ${\sup _{\left \| x \right \| = 1}}\{ {\left \| {Tx} \right \|^2} - |(Tx,x){|^2}\} \geqslant R_T^2$, where ${R_T}$ is the radius of the smallest disc containing the spectrum of T. An example is given, which shows that we cannot expect equality for ${G_1}$ operators in general, contrary to a conjecture of Istrǎţescu.


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Keywords: Spectrum, <IMG WIDTH="30" HEIGHT="38" ALIGN="MIDDLE" BORDER="0" SRC="images/img1.gif" ALT="${G_1}$"> operator
Article copyright: © Copyright 1980 American Mathematical Society