If
f is an analytic function bounded on a convex domain of the
complex plane and
A a square matrix whose spectrum is included in this
domain, the function
f(
A) is well defined. In this paper we study bounds for
||
f(
A)|| uniform with respect to the functions
f bounded by 1, and uniform
with respect to the matrices
A whose the numerical ranges are included in the
domain. We show that these bounds are attained and give explicit formulae
in some 2-dimensional cases.
Mathematics Subject Classification (2000). 47A12
Keywords. Numerical range