Symbolic calculus of operators with unit numerical radius Original Research Article

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 supzless-than-or-equals, slant1p(z)less-than-or-equals, slant1 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 Andôimagefor the operator norm and the estimateimagefor the numerical radius.