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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Operators satisfying certain growth conditions


Authors: S. M. Patel and B. C. Gupta
Journal: Proc. Amer. Math. Soc. 53 (1975), 341-346
MSC: Primary 47A65
MathSciNet review: 0385617
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Abstract: Let $ T$ be an operator on a complex Hilbert space $ H$. Some growth conditions on operator radius of the resolvent of $ T$ are studied. Moreover, it is shown that the conjecture, due to V. Istrătescu, that for operators $ T$ satisfying growth condition $ ({{\text{G}}_1})$

$\displaystyle \mathop {\sup }\limits_{\vert\vert x\vert\vert = 1} \;\{ \vert\vert Tx\vert{\vert^2} - \vert(Tx,\;x){\vert^2}\} = R_T^2,$

where $ {R_T}$ is the radius of the smallest circular disk containing the spectrum $ \sigma (T)$, turns out to be false.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1975-0385617-0
PII: S 0002-9939(1975)0385617-0
Keywords: Hilbert space operators, $ \rho $-dilation, operator radii, transloid operators, selfadjoint and unitary operators, operators satisfying the growth condition $ ({{\text{G}}_1})$
Article copyright: © Copyright 1975 American Mathematical Society