Abstract. An eigenvalue criterion for hypercyclicity due to the first author is improved. As a consequence, some new sufficient conditions for a sequence of infinite order linear differential operators to be hypercyclic on the space of holomorphic functions on certain domains of C N are shown. Moreover, several necessary conditions are furnished. The equicontinuity of a family of operators as above is also studied, and it is characterized if the domain is C N . The results obtained extend or improve earlier work of several authors. Introduction, notation and preliminary results.Throughout this paper we denote by N the set of positive integers, by R the real line, by C the field of complex numbers, and by N 0 the set N 0 = N ∪ {0}. Let X, Y be two linear topological spaces, T i : X → Y (i ∈ I := an arbitrary index set) a family of continuous linear mappings, and x ∈ X. Then x is said to be hypercyclic or universal for (T i ) whenever its orbit {T i x : i ∈ I} under (T i ) is dense in Y . The family (T i ) is called hypercyclic whenever it has a hypercyclic vector. Note that if (T i ) is hypercyclic then it is not equicontinuous, but the converse is false in general. In the case I = N, it is clear that, in order that a sequence (T n ) can be hypercyclic, Y must be separable. If T : X → X is an operator (= continuous linear selfmapping) on X, then a vector x ∈ X is said to be hypercyclic for T if it is hypercyclic for the sequence (T n ) of iterates of T , i.e., T n = T • . . . • T (n-fold). The operator T is hypercyclic when there is a hypercyclic vector for T . The symbols HC(T ) and HC((T i )) will denote, respectively, the set of hypercyclic vectors of an operator T and of a family T i : X → Y (i ∈ I) of continuous linear mappings. In the last two

Inspired by a statement of W. Luh asserting the existence of entire functions having together with all their derivatives and antiderivatives some kind of additive universality or multiplicative universality on certain compact subsets of the complex plane or of, respectively, the punctured complex plane, we introduce in this paper the new concept of U-operators, which are defined on the space of entire functions. Concrete examples, including differential and antidifferential operators, composition, multiplication and shift operators, are studied. A result due to Luh, Martirosian and Miiller about the existence of universal entire functions with gap power series is also strengthened.2000 Mathematics subject classification: primary 30E10; secondary 47A16, 47B33, 47B38, 47E05, 47G10.

Several necessary and sufficient conditions for a sequence of infinite order differential linear operators on spaces of holomorphic functions on a domain of the complex plane to be supercyclic or c-hypercyclic are given in this paper, so completing earlier work of the authors on hypercyclicity, which in turn extended Birkhoff-MacLane-Godefroy-Shapiro's theorems. A new, general eigenvalue criterium for supercyclicity is also provided.