An infinite polynomially nonlinear system of equations (Q1919575)

The author proves the existence of the solution of the following system of polynomial equations \[ \sum_{m=0}^{\infty }a_{m,n}^{k+1}=\delta_{nk}, \tag{1} \] where \(k\) is an exponent, \(n,m\) are nonnegative integers, \(a_{m,n}\) for every \(n,m\) is an unknown complex number and \(\delta_{nk}\) is the Kronecker delta. Using the solution of system (1) it is possible to solve some problems in nonlinear functional analysis, in particular, a nonlinear operator equation on \(C(K)\), where \(K\) is a compact metric space which is the continuous image of the unit interval \([0,1]\) and which contains a closed subspace homeomorphic to \([0,1]\). More exactly, for a given nonnegative integer \(n\) there exist bounded linear operators \(A, B\) on \(C(K)\), \(|A|\leq 1\), such that \[ B(1-Af)^{-1}=f^n \tag{2} \] for all \(f\in C(K)\), \(|f|\leq 1\). Equation (2) is a paradoxical, typically infinite dimensional result, since it is not possible to find complex numbers \(a\) and \(b\) such that \(a(1-bz)^{-1}=z^n\) would hold for every complex \(z\), \(|z|<1\).