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Numerical peak points and numerical Šilov boundary for holomorphic functions. (English) Zbl 1178.46033

This paper deals with the concept of numerical boundary for subspaces of C b (B E :E), the space of continuous functions from the unit ball B E of a Banach space E into the space E, 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 hC b (B E :E) is v(h)=sup{|x * (h(x))|:(x,x * )Π(E)}, where

Π(E)={(x,x * )X×X * :x * =x=x * (x)=1}·

A subset BΠ(E) is a numerical boundary for 𝒜C b (B E :E) provided that

sup (x,x * )B |x * (h(x))|=v(h)

for every h𝒜. B is said to be the numerical Silov boundary for 𝒜 if it is the minimal (·×w * )-closed numerical boundary for 𝒜. For a complex Banach space E, let 𝒜 (B E :E) denote the subspace of space of C b (B E :E) of those bounded continuous functions from B E into E which are holomorphic on the open unit ball.

The main result of the paper is that the set

{(x,x * )Π( 1 ):|x * (e n )|=1n}

is the numerical Silov boundary for 𝒜 (B 1 : 1 ).

Numerical peak points and numerical strong-peak points for 𝒜 (B C(K) :C(K)) are also studied.

MSC:
46E40Spaces of vector- and operator-valued functions
47A12Numerical range and numerical radius of linear operators
46E50Spaces of differentiable or holomorphic functions on infinite-dimensional spaces