The Fréchet Schwartz Algebra of Uniformly Convergent Dirichlet Series

Abstract

The algebra of all Dirichlet series that are uniformly convergent in the half-plane of complex numbers with positive real part is investigated. 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 aQ-algebra. Composition operators on this space are also studied.

The differentiation operator in the space of uniformly convergent Dirichlet series

“…Proceeding as in the proof of Theorem 2.2 in [5], we get lim k→∞ a k n = a 0 n and lim k→∞ a k n γ n = b 0 n for each n. Therefore γ n a 0 n = b 0 n and T has closed graph. Since H ∞ +,0 is a Fréchet space, the closed graph theorem implies that T is continuous.…”

“…The Fréchet space H ∞ + of Dirichlet series f (s) = ∞ n=1 a n n −s which are uniformly convergent on the half-planes C ε := {s ∈ C | Re s > ε} for each ε > 0 was investigated by the author in [5]. When endowed with its natural metrizable locally convex topology, it is Schwartz, not nuclear, has a Schauder basis and contains isomorphically the space H(D m ) of analytic functions on the open unit polydisc D m for each m ∈ N. Moreover, this space is a multiplicatively convex Fréchet algebra for the pointwise product.…”

“…As in [5], we denote by H ∞ + the space of all analytic functions on the half-plane C 0 which are bounded on C ε for each ε > 0 and that can be represented as a convergent Dirichlet series…”

“…a complete metrizable locally convex space. It was proved in [5] that H ∞ + is Schwartz, non-nuclear and the Dirichlet monomials e n (s) = n −s , n ∈ N are a Schauder basis of the space. In fact, [20,Theorem 6.1.1] implies that all coefficient functionals u n : H ∞ + → C, f → a n of the Dirichlet monomials e n are continuous.…”

“…If X is Montel (i.e., each bounded set is relatively compact), then T is compact if and only if it is bounded. Since H ∞ + is a Fréchet-Schwartz space [5], hence a Montel space, there is no distinction between being compact or bounded for a continuous linear operator on H ∞ + .…”