Operators satisfying certain growth conditions

Patel, S. M.; Gupta, B. C.

doi:10.1090/S0002-9939-1975-0385617-0

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
DOI: https://doi.org/10.1090/S0002-9939-1975-0385617-0
MathSciNet review: 0385617
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Abstract: Let be an operator on a complex Hilbert space . Some growth conditions on operator radius of the resolvent of are studied. Moreover, it is shown that the conjecture, due to V. Istrătescu, that for operators 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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