Universal functions on complex special linear groups. (Q2751614)

Let an entire function \(F(\zeta)\) be universal, i.e. for any entire function \(f(\zeta)\) there exists a sequence \(\{a_n\}\) of complex numbers such that NEWLINE\[NEWLINEf(\zeta)= \lim_{n\to \infty}F (\zeta+a_n)NEWLINE\]NEWLINE where the convergence is uniformly on compact sets.NEWLINENEWLINENEWLINEThe author considers universal functions of several variables.NEWLINENEWLINENEWLINEThe main result of this paper isNEWLINENEWLINENEWLINETheorem 4. Let \(S\) be the set of all square matrices of degree \(n\) with complex coefficients such that their determinants are equal to 1.NEWLINENEWLINENEWLINEThere exist a holomorphic function \(F\) on \(S\) and \(C\in S\) such that for any compact set \(K\) with connected complement in \(S\), for any holomorphic on \(K\) function and any \(\varepsilon >0\) NEWLINE\[NEWLINE\max_{Z\in K}\bigl|F(CZ)-f(Z) \bigr|<\varepsilon.NEWLINE\]NEWLINENEWLINENEWLINEFor the entire collection see [Zbl 0964.00041].