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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 𝒜𝒞 b (Ω). This is a subset of Ω that is a norming set for 𝒜. We introduce the concept of numerical boundary. For a Banach space X, a subset BΠ(X) is a numerical boundary for a subspace 𝒜𝒞 b (B X ,X) 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(K) and d * (w,1), the predual of the Lorentz sequence space d(w,1). 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:
46B04Isometric theory of Banach spaces
46E15Banach spaces of continuous, differentiable or analytic functions
47A12Numerical range and numerical radius of linear operators
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