Abstract. We show that an invertible operator $A$ is unitary if the numerical radius satisfies $w(A) \leqq 1$ and $w(A^{-1}) \leqq 1$.

AMS Subject Classification (1991): 47A12, 47A63

Keyword(s): numerical radius, unitary operator, numerical range

Received August 5, 2009, and in revised form August 28, 2009. (Registered under 93/2009.)