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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 ${C}_{b}\left({B}_{E}:E\right)$, 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 $h\in {C}_{b}\left({B}_{E}:E\right)$ is $v\left(h\right)=sup\left\{|{x}^{*}\left(h\left(x\right)\right)|:\left(x,{x}^{*}\right)\in {\Pi }\left(E\right)\right\}$, where

${\Pi }\left(E\right)=\left\{\left(x,{x}^{*}\right)\in X×{X}^{*}:\parallel {x}^{*}\parallel =\parallel x\parallel ={x}^{*}\left(x\right)=1\right\}·$

A subset $B\subset {\Pi }\left(E\right)$ is a numerical boundary for $𝒜\subset {C}_{b}\left({B}_{E}:E\right)$ provided that

$\underset{\left(x,{x}^{*}\right)\in B}{sup}|{x}^{*}\left(h\left(x\right)\right)|=v\left(h\right)$

for every $h\in 𝒜$. $B$ is said to be the numerical Silov boundary for $𝒜$ if it is the minimal $\left(\parallel ·\parallel ×{w}^{*}\right)$-closed numerical boundary for $𝒜$. For a complex Banach space $E$, let ${𝒜}_{\infty }\left({B}_{E}:E\right)$ denote the subspace of space of ${C}_{b}\left({B}_{E}:E\right)$ 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

$\left\{\left(x,{x}^{*}\right)\in {\Pi }\left({\ell }_{1}\right):|{x}^{*}\left({e}_{n}\right)|=1\phantom{\rule{4pt}{0ex}}\forall n\in ℕ\right\}$

is the numerical Silov boundary for ${𝒜}_{\infty }\left({B}_{{\ell }_{1}}:{\ell }_{1}\right)$.

Numerical peak points and numerical strong-peak points for ${𝒜}_{\infty }\left({B}_{C\left(K\right)}:C\left(K\right)\right)$ are also studied.

##### MSC:
 46E40 Spaces of vector- and operator-valued functions 47A12 Numerical range and numerical radius of linear operators 46E50 Spaces of differentiable or holomorphic functions on infinite-dimensional spaces