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Numerical boundaries for some classical Banach spaces. (English) Zbl 1167.46008
Summary: J. Globevnik [Math. Proc. Camb. Philos. Soc. 85, 291–303 (1979; Zbl 0395.46040)] gave the definition of boundary for a subspace $𝒜\subset {𝒞}_{b}\left({\Omega }\right)$. This is a subset of ${\Omega }$ that is a norming set for $𝒜$. We introduce the concept of numerical boundary. For a Banach space $X$, a subset $B\subset {\Pi }\left(X\right)$ is a numerical boundary for a subspace $𝒜\subset {𝒞}_{b}\left({B}_{X},X\right)$ if the numerical radius of $f$ is the supremum of the modulus of all the evaluations of $f$ at $B$, for every $f$ in $𝒜$. We give examples of numerical boundaries for the complex spaces $X={c}_{0}$, $𝒞\left(K\right)$ and ${d}_{*}\left(w,1\right)$, the predual of the Lorentz sequence space $d\left(w,1\right)$. In all these cases (if $K$ is infinite), we show that there are closed and disjoint numerical boundaries for the space of the functions from ${B}_{X}$ to $X$ which are uniformly continuous and holomorphic on the open unit ball and there is no minimal closed numerical boundary. In the case of ${c}_{0}$, we characterize the numerical boundaries for that space of holomorphic functions.

##### MSC:
 46B04 Isometric theory of Banach spaces 46E15 Banach spaces of continuous, differentiable or analytic functions 47A12 Numerical range and numerical radius of linear operators
##### Keywords:
holomorphic function; boundary; Šilov boundary