Inner functions of numerical contractions

We prove that, for a function f in H^∞ of the unit disc with {norm of matrix} f {norm of matrix}_∞ ≤ 1, the existence of an operator T on a complex Hilbert space H with its numerical radius at most one and with {norm of matrix} f (T) x {norm of matrix} = 2 for some unit vector x in H is equivalent to that f be an inner function with f (0) = 0. This confirms a conjecture of Drury [S.W. Drury, Symbolic calculus of operators with unit numerical radius, Linear Algebra Appl. 428 (2008) 2061-2069]. Moreover, we also show that any operator T satisfying the above conditions has a direct summand similar to the compression of the shift S (φ{symbol}), where φ{symbol} (z) = zf (z) for | z | < 1. This generalizes the result of Williams and Crimmins [J.P. Williams, T. Crimmins, On the numerical radius of a linear operator, Amer. Math. Monthly 74 (1967) 832-833] for f (z) = z and of Crabb [M.J. Crabb, The powers of an operator of numerical radius one, Michigan Math. J. 18 (1971) 253-256] for f (z) = zⁿ (n ≥ 2).