The paper deals with an extension theorem by Costakis and Vlachou on simultaneous approximation for holomorphic function to the setting of Dirichlet series, which are absolutely convergent in the right half of the complex plane. The derivation operator used in the analytic case is substituted by a weighted backward shift operator in the Dirichlet case. We show the similarities and extensions in comparing both results. Several density results are proved that finally lead to the main theorem on simultaneous approximation.

We express the structure of some positive polynomials in several variables, as squares of rational function with universal denominators, using functional analysis methods. ᮊ

Abstract. We study universal Dirichlet series with respect to overconvergence, which are absolutely convergent in the right half of the complex plane. In particular we obtain estimates on the growth of their coefficients. We can then compare several classes of universal Dirichlet series.

The point source of this work is Seleznev's theorem which asserts the existence of a power series which satisfies universal approximation properties in C * . The paper deals with a strengthened version of this result. We establish a double approximation theorem on formal power series using a weighted backward shift operator. Moreover we give strong conditions that guarantee the existence of common universal series of an uncountable family of weighted backward shift with respect to the simultaneous approximation. Finally we obtain results on admissible growth of universal formal power series. We especially prove that you cannot control the defect of analyticity of such a series even if there exist universal series in the well-known intersection of formal Gevrey classes.

We introduce the notion of (joint) formal normality for a collection of unbounded linear operators on a separable Hilbert space H which is, in some sense, a natural generalization of the notion of formal normality for a single operator. We give some relations between this new notion and (joint) subnormality and hyponormality. We adapt, in particular, a proof of Stochel and Szafraniec to give necessary and sufficient conditions for a tuple of unbounded operators with invariant domain to be (jointly) subnormal. (2000). Primary 47B20; Secondary 47A20. Mathematics Subject Classification