Abstract. Let $ {\cal B}({\cmss H})$ be the algebra of all bounded linear operators on a complex separable Hilbert space $ {\cmss H}$, and denote by $\gamma(T)$ the reduced minimum modulus of $T\in{\cal B}({\cmss H})$. Mbekhta [Mbekhta2007] conjectured that a surjective linear map $\phi\colon {\cal B}({\cmss H}) \rightarrow{\cal B}({\cmss H}) $ verifying $\gamma(T) = \gamma(\phi(T) ) $ for every $T\in{\cal B}({\cmss H})$ if and only if $\phi $ takes one of the following forms: $\phi(T) = UTV $ for every $T\in{\cal B}({\cmss H})$, or $\phi(T) = UT^{tr}V $ for every $T\in{\cal B}({\cmss H})$, where $U\in{\cal B}({\cmss H})$ and $ V\in{\cal B}({\cmss H}) $ are unitary operators. We answer in the affirmative a problem raised by the conjecture.

AMS Subject Classification (1991): 47B48, 47A30

Keyword(s): reduced minimum modulus, generalized spectrum, unitary operator, linear preservers

Received October 16, 2007, and in revised form December 18, 2007. (Registered under 6050/2009.)