room: A1-33, 10:30–12:00

José Bonet (UPV Universitat Politècnica de València)
Some questions about spaces of Dirichlet series

Summary. In the first part of this lecture, which is of survey type, we recall several results about Bohr problem concerning the largest possible strip on which a Dirichlet series of complex numbers converges uniformly but not absolutely. We will explain classical work by Harald Bohr (1913), Bohnenblust and Hille (1931), and recent one by Boas, Defant, Frerick, García, Khavinson, Maestre, Ortega-Cerdà, Ounaies and Seip, among others.

In the second part we will report about our recent work concerning the following questions.

(1) Defant, García, Maestre, Pérez-García investigated in 2008 Bohr’s problem for Dirichlet series with values on a Banach space. They showed that the largest possible strip on which a Dirichlet series with coefficients in a Banach space converges uniformly but not absolutely depends on the geometry of the Banach space, in particular on its cotype.

The abscissas of convergence, uniform convergence and absolute convergence of vector valued Dirichlet series with respect to the original topology and with respect to the weak topology of a locally convex space \(X\) are compared. The relation of their coincidence with geometric or topological properties of the underlying space \(X\) is investigated. Cotype in the context of Banach spaces, and nuclearity and certain topological invariants for Fréchet spaces play a relevant role.

(2) Motivated by a classical result of Bohr that the abscissa of boundedness and the abscissa of uniform convergence coincide for a Dirichlet series and by an improved Montel principle due to Bayart in 2002, we investigate the algebra of all Dirichlet series that are uniformly convergent in the half-plane of complex numbers with positive real part. When it is endowed with its natural locally convex topology it is a non-nuclear, Fréchet Schwartz space with basis. Moreover, it is a locally multiplicative algebra but not a \(Q\)-algebra.

Coffee and the at 10:00.