Inner functions of numerical contractions

We prove that, for a function f in H∞ of the unit disc with ‖f‖∞⩽1, the existence of an operator T on a complex Hilbert space H with its numerical radius at most one and with ‖f(T)x‖=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), where ϕ(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).