This paper deals with the concept of numerical boundary for subspaces of , the space of continuous functions from the unit ball of a Banach space into the space , which was introduced in [M. D. Acosta and S. G. Kim, J. Math. Anal. Appl. 350, No. 2, 694–707 (2009; Zbl 1167.46008)]. Let us recall that the numerical radius of a function is , where
A subset is a numerical boundary for provided that
for every . is said to be the numerical Silov boundary for if it is the minimal -closed numerical boundary for . For a complex Banach space , let denote the subspace of space of of those bounded continuous functions from into which are holomorphic on the open unit ball.
The main result of the paper is that the set
is the numerical Silov boundary for .
Numerical peak points and numerical strong-peak points for are also studied.