Operators satisfying certain growth conditions

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

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

Operators satisfying certain growth conditions
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by S. M. Patel and B. C. Gupta PDF

Proc. Amer. Math. Soc. 53 (1975), 341-346 Request permission

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 $({{\text {G}}_1})$ \[ \sup \limits _{||x|| = 1} \;\{ ||Tx|{|^2} - |(Tx,\;x){|^2}\} = R_T^2,\] 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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