Bohr’s strip for vector valued Dirichlet series

Abstract

Bohr showed that the width of the strip (in the complex plane) on which a given Dirichlet series a n /n s , s ∈ C, converges uniformly but not absolutely, is at most 1/2, and Bohnenblust-Hille that this bound in general is optimal. We prove that for a given infinite dimensional Banach space Y the width of Bohr's strip for a Dirichlet series with coefficients a n in Y is bounded by 1 − 1/ Cot(Y ), where Cot(Y ) denotes the optimal cotype of Y . This estimate even turns out to be optimal, and hence leads to a new characterization of cotype in terms of vector valued Dirichlet series. Mathematics Subject Classification (2000)Primary 32A05; Secondary 46B07 · 46B09 · 46G20 The first, second and third authors were supported by MEC and FEDER Project MTM2005-08210. A. Defant

Estimates for vector valued Dirichlet polynomials

“…But if X is infinite dimensional these two numbers coincide and depend only on the optimal cotype of X. More precisely, the main result from [11] shows that for any infinite dimensional Banach space X (see Section 2 for the defintinion of cot(X))…”

“…The study of the width of Bohr's strips for such objects was initiated in [11] and continued in [15]. Given a operator v : X → Y between two Banach spaces we define the number…”

Convergence of Dirichlet polynomials in Banach spaces

“…However, if X is infinite dimensional, then both numbers coincide and depend only on the geometry of the underlying Banach space X. More precisely, the main result from [11] states that for an infinite dimensional Banach space X and for each m we have T (X) = T m (X) = 1 − 1/ cot(X) , where cot(X) is the optimal cotype of X (see below for definitions). For special Banach spaces this optimal cotype is computable which e.g.…”

“…Based on our results from [13] we continue the recent study of ordinary Dirichlet series from [11]. Let X be some Banach space and m ∈ N. We call a series n a n 1 n s , s ∈ C, a Dirichlet series in X if all its coefficients a n belong to X.…”

“…More recently, in [11] a similar question for Dirichlet series n a n 1 n s , s ∈ C, with coefficients a n in a fixed Banach space X is addressed. Clearly, the definition of the maximal width T (X) and T m (X) then again follows the same lines, but considerably different phenomena occur.…”