Large vector spaces generated by bounded universal functions

We provide a general framework to construct large vector spaces generated by bounded universal functions. We provide necessary and sufficient conditions on a sequence of automorphisms phi(n) of a domain Omega in C-N for there to exist a normclosed infinite-dimensional vector space V generated by functions in B = {f is an element of H-infinity (Omega) vertical bar parallel to f parallel to(infinity) <= 1} such that V is locally uniformly dense in H-infinity (Omega), V is linearly isometric with l(1), and every f is an element of V with parallel to f parallel to = 1 is universal for the composition operators C-phi n : f bar right arrow f omicron phi(n), by which we mean the set {f omicron phi(n) vertical bar n >= 1} is locally uniformly dense in B. In the case when Omega is a bounded symmetric domain in C-N, we provide sufficient conditions on a sequence of holomorphic self-maps phi(n) of Omega for there to exist a vector space V generated by a locally uniformly dense set of inner functions such that V has each of the properties mentioned above. (C) 2020 Elsevier Inc. All rights reserved.