Characterization of the class of unitary operators by operator inequalities

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 .

Keywords:

• invertible operator
• unitary operator
• injective norm
• operator inequality

AMS Subject Classification::

• 47B47

Acknowledgement

The author would like to thank the referee for this careful reading of this article and his useful comments.