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On the relation between number ranges of the characteristics for a Banach space and its subspaces. (Russian) Zbl 0632.46013

Let X be a Banach space, \(X^*\) the topological dual of X, F a total linear subset in \(X^*\). The number \(r(F)=\inf \{\sup \{| f(x)|:\) \(f\in F\), \(\| f\| \leq 1\}\), \(x\in X\), \(\| x\| =1\}\) is said to be the characteristic of F. The set \(R(X)=\{\lambda\); \(\exists F\subset X^*\), \(r(F)=\lambda \}\) is said to be the numerical range of the characteristic. In B. V. Godun, M. I. Kadets [ibid. 29, 25-31 (1978; Zbl 0439.46011)] some properties of R(X) have been considered. A family of Banach spaces \(X_ n\), \(n\in {\mathbb{N}}\) is given such that \(\forall X_ n\), \(\exists Y\subset X_ n\) satisfying \(R(Y)=[0,1]\) and \(R(X_ n)=[0,(2n+1)/(4n+1)]\). If \(r(F)>0\) then F is said to be norming for X. Let \(Y\subset X\), and \(R: X^*\to Y^*\) the canonical map. If \(G\subset Y^*\) is norming for Y then \(F=R^{-1}G\) is norming for X and r(F)\(\geq r(G)/3\) [W. J. Davis, W. B. Johnson, Isr. J. Math. 14, 353-367 (1973; Zbl 0273.46009)].
The author’s theorem: let \(X=Y\dot +Z\) be a Banach space as a direct sum of its subspaces, \(\| P\| =1\), where P denotes the projection onto Y parallel to Z, \(\| I-P\| =1\). Then for each \(G\subset Y^*\) norming for Y it holds r(F)\(\geq r(G)/2\). An example is given showing that the estimate r(F)\(\geq r(G)/3\) cannot be improved.
Reviewer: S.Suljagic

MSC:

46B10 Duality and reflexivity in normed linear and Banach spaces
46B20 Geometry and structure of normed linear spaces
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