Abstract
Let ,
and
be the C*-algebra of all bounded linear operators acting on a complex Hilbert space H, the set of all invertible elements in
and the class of all unitary operators in
, respectively. In this note, we shall show that if
, then the injective norm of S ⊗ S −1 + S −1 ⊗ S in the tensor product space
attains its minimal value 2 if and only if S is normal and satisfies the condition
for every λ, μ in the spectrum σ(S) of S. Finally, it is shown that if
then the inequality ‖SXS −1 + S −1
XS‖ ≤ 2‖X‖ holds for all X in
if and only if
.
AMS Subject Classification::
Acknowledgement
The author would like to thank the referee for this careful reading of this article and his useful comments.