A characterization of Banach spaces with nonzero Bohr radius
type: Item
instance of: scholarly article
title: A characterization of Banach spaces with nonzero Bohr radius
zbMATH Open document ID: 1468.30007
author: Bappaditya Bhowmik
author: Nilanjan Das
published in: Archiv der Mathematik
publication date: 2021-05-05T00:00:00Z
review text: 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)].
Mathematics Subject Classification ID: 30B10
Mathematics Subject Classification ID: 32K12
Mathematics Subject Classification ID: 46E40
zbMATH DE Number: 7343934
zbMATH Keywords: Bohr radius
zbMATH Keywords: uniform convexity of Banach spaces
reviewed by: Oscar Blasco
MaRDI profile type: Publication
full work available at URL: s00013-020-01568-8
OpenAlex ID: W3126314302
cites work: Bohr property of bases in the space of entire functions and its generalizations
cites work: The Bohr Radius of a Banach Space
cites work: The \(p\)-Bohr radius of a Banach space
cites work: Bohr’s power series theorem in several variables
cites work: The complex convexity of quasi-normed linear spaces
cites work: Bohr's strip for vector valued Dirichlet series
cites work: Banach Algebras Satisfying the Non-Unital Von Neumann Inequality
cites work: Q3535672
cites work: On Complex Strict and Uniform Convexity
cites work: Bohr's theorem for holomorphic mappings with values in homogeneous balls
cites work: Bohr’s inequality for uniform algebras
cites work: Bohr inequalities for free holomorphic functions on polyballs
cites work: The Strong Maximum Modulus Theorem for Analytic Functions into a Banach Space
DOI: 10.1007/S00013-020-01568-8
description: scientific article; zbMATH DE number 7343934