Let \mathcal H be a complex Hilbert space, with no separability assumption, and let A\in\mathcal B(\mathcal H). Suppose first that \mathcal H\ne\{0\}, and put
W(A)=\{\langle x,Ax\rangle:x\in\mathcal H,\ \|x\|=1\},
\qquad K=\overline{W(A)}.
Then K is compact and convex and contains \spec(A). For every positive integer m, every nonnegative integer d, and every matrix polynomial P(z)=\sum_{k=0}^d B_kz^k with B_k\in M_m(\C), the operator P[A]=\sum_{k=0}^d A^k\otimes B_k on \mathcal H\otimes\C^m satisfies
\|P[A]\|\le 2\sup_{z\in W(A)}\|P(z)\|
=2\max_{z\in K}\|P(z)\|.
More generally, if U is an open neighborhood of K and F=(f_{ij}):U\to M_m(\C) is holomorphic, its entrywise holomorphic evaluation, in the same base-first tensor convention, satisfies
F[A]:=\sum_{i,j=1}^m f_{ij}(A)\otimes E_{ij},
\qquad \|F[A]\|\le2\max_{z\in K}\|F(z)\|.
Here E_{ij} are the standard matrix units and f_{ij}(A) is given by the holomorphic functional calculus. In particular, K is a complete 2-spectral set for A: the same estimate holds at every finite coefficient size for every matrix-valued rational function whose poles lie outside K, with evaluation by the rational functional calculus. The constant two is optimal uniformly over all these choices.
If \mathcal H=\{0\}, every evaluation is the zero operator. With the supremum over the empty numerical range defined to be zero, the corresponding supremum-norm inequalities are 0\le0; the compact spectral-set assertion above is stated for nonzero \mathcal H.