Universal Functions on Complex General Linear Groups Original Research Article

In 1929, Birkhoff proved the existence of an entire function F on C with the property that for any entire function f there exists a sequence {ak} of complex numbers such that {F(ζ+ak)} converges to f (ζ) uniformly on compact sets. Luh proved a variant of Birkhoffʹs theorem and the second author proved a theorem analogous to that of Luh for the multiplicative group C*. In this paper extensions of the above results to the multi-dimensional case are proved. Let M(n, C) be the set of all square matrices of degree n with complex coefficients, and let G=GL(n, C) be the general linear group of degree n over C. We denote by A(G) the set of all holomorphic functions on G. Similarly, we define A(C). Let K be the A(G)-hull of a compact set K in G. Finally we denote by B(G) the set of all compact subsets K of G with K=K such that there exists a holomorphic function f on M(n, C) with f(0)∉(f(K)), where (f(K)) is the A(C)-hull of f(K). Our main result is the following. There exists a holomorphic function F on G such that for any K∈B(G), for any function f holomorphic in some neighbourhood of K, and for any ε>0, there exists C∈G with maxZ∈K |F(CZ)−f(Z)|<ε.