Symbolic calculus of operators with unit numerical radius

Let T be a linear operator on a complex Hilbert space with numerical radius bounded by one. We study the norm and numerical range of p(T) where p   is a disk algebra function satisfying sup|z|⩽1|p(z)|⩽1sup|z|⩽1|p(z)|⩽1 and p(0) is known. As corollaries we are able to establish for p an arbitrary complex polynomial the known estimate due to Okubo and Andô‖p(T)‖⩽2sup|z|⩽1⩽2sup|z|⩽1|p(z)|for the operator norm and the estimatew(p(T))⩽54sup|z|⩽1|p(z)|for the numerical radius.