Variable Compleja, eSpacios de F unciones y ⊙peradores enTre ellos Problemas de Investigaci´ on C F un Sp⊙T Research Problems Section Complex variables, F unction Spaces and ⊙perators between Them APPROXIMATION BY EULER OPERATORS OUTSIDE THE ORIGIN L. BERNAL-GONZ ´ ALEZ, M.C. CALDER ´ ON-MORENO, AND J.A. PRADO-BASSAS Abstract. We present an open problem concerning the universality of a special class of diagonal operators acting on the space of entire functions. Let G be a domain in the complex plane C, with 0 ∈ G, and X be a topological vector space such that X ⊂ H (G), where H (G) denotes the space of all holomorphic functions on G. We say that a (linear, continuous) operator T : X → X is a diagonal operator (or a coefficient multiplier) whenever there exists a sequence α =(a n ) n≥0 ⊂ C such that for every f ∈ X with f (z )= ∞  n=0 f n z n around the origin, we have Tf (z )= ∞  n=0 a n f n z n around the origin. We denote T = T α . For instance, if G = C and α =(a n ) n≥0 ⊂ C with lim sup n→∞ |a n | 1/n < +∞, then T α is a diagonal operator on H (C). A remarkable instance of these operators is the class of Euler operators, whose definition will be recalled later. Before going on, we recall three degrees of approximation by operators in a topological vector space X . An operator T : X → X is said to be cyclic (supercyclic, hypercyclic, resp.) if there is a vector x 0 ∈ X such that the linear span span{T n x 0 } n≥0 of the orbit of x 0 under T (the projective orbit {λT n x 0 : λ ∈ C,n ≥ 0}, the orbit itself {T n x 0 : n ≥ 0} of x 0 under T , 2000 Mathematics Subject Classification. 30E10, 47A16, 47B38. Key words and phrases. Diagonal operators, Euler operators, hypercyclic operators, exponential type. Research supported in part by the Plan Andaluz de Investigaci´on de la Junta de An- daluc´ ıa FQM-127 and by MCYT Grants MTM2006-13997-C02-01 and MTM2006-26627-E. 1