A characterization of Banach spaces with nonzero Bohr radius
The authors give a characterization of Banach spaces satisfying that certain modification of the Bohr radius hold true. In the work by the reviewer [Oper. Theory: Adv. Appl. 201, 59--64 (2010; Zbl 1248.46030)] the quantity \(R_{p,q}(X)\) associated to a complex Banach space and values \(1\le p,q<\infty\), given by \[R_{p,q}(X)=\inf\left\{\sup\Big\{r>0: \|x_0\|^p+ \Big(\sum_{n=1}^\infty \|x_n\|r^n\Big)^q\le 1\Big\}: \Big\|\sum_{n=0}^\infty x_nz^n\Big\|_{H^\infty(\mathbb D, X)}\le 1\right\},\] was introduced. A variation of this quantity \[r_p(X)=\sup\left\{r>0:\Big(\sum_{n=1}^\infty \|x_n\|^pr^n\Big)^{1/p}\le \Big\|\sum_{n=0}^\infty x_nz^n\Big\|_{H^\infty(\mathbb D, X)}\right\}\] was also considered [the reviewer, Collect. Math. 68, No. 1, 87--100 (2017; Zbl 1371.46013)]. \textit{H. Bohr} showed that \(r_1(\mathbb C)=R_{1,1}(\mathbb C)=1/3\) [Proc. Lond. Math. Soc. (2) 13, 1--5 (1914; JFM 44.0289.01)]. Since then many work has been done on the so-called Bohr radius. Much later it was observed in the mentioned papers that for vector-valued analytic functions in general the values \(R_{p,q}(X)\) and \(r_p(X)\) need not be positive even for Banach spaces of dimension bigger than 1 in some cases. In this paper the authors characterize the case \(R_{p,q}(X)>0\) in terms of the existence of a constant \(C\) such that \(\Omega_X(\delta)\le C( (1+\delta)^q-(1+\delta)^{q-p})^{1/q}\) for all \(\delta>0\) where \(\Omega\) stands for the modulus of complex convexity given by \(\Omega(\delta)=\sup \{ \|y\|: \|x\|=1, \sup_{|z|<1} \|x+zy\|\le 1+\delta\}\), introduced by \textit{J. Globevnik} [Proc. Am. Math. Soc. 47, 175--178 (1975; Zbl 0307.46015)]. Since \(r_p(X)>0\) iff \(R_{p,p}(X)>0\) then this result completes the characterization of spaces with \(r_p(X)>0\) for \(p\ge 2\) as those which are \(p\)-uniformly \(\mathbb C\)-convex due to [Zbl 1371.46013)].
scientific article; zbMATH DE number 7343934
scientific article