Upper and lower bounds on norms of functions of matrices

Given an n by n matrix A, we look for a set S in the complex plane and positive scalars m and M such that for all functions p bounded and analytic on S and throughout a neighborhood of each eigenvalue of A, the inequalitiesm·inf{‖f‖L∞(S):f(A)=p(A)}⩽‖p(A)‖⩽M·inf{‖f‖L∞(S):f(A)=p(A)}m·inf{‖f‖L∞(S):f(A)=p(A)}⩽‖p(A)‖⩽M·inf{‖f‖L∞(S):f(A)=p(A)}hold. We show that for 2 by 2 matrices, if S   is the field of values, then one can take m=1m=1 and M=2M=2. We show that for a perturbed Jordan block – a matrix A that is an n by n   Jordan block with eigenvalue 0 except that its (n,1)(n,1)-entry is νν, with |ν|∈(0,1)|ν|∈(0,1) – if S   is the unit disk, then m=M=1m=M=1. We argue, however, that, in general, due to the behavior of minimal-norm interpolating functions, it may be very difficult or impossible to find such a set S   for which the ratio M/mM/m is of moderate size.