Subsymmetric functions on Banach spaces with subsymmetric bases

Abstract

Properties of subsymmetric polynomials, analytic functions and some their generalizations on Banach spaces with subsymmetric bases are considered. We prove that if a polynomial on a complex infinite-dimensional Banach space $X$ has a subsymmetric set of zeros, then it is subsymmetric. From here we deduce that the algebra $\mathcal{P}_{\mathfrak{S}}(X)$ of subsymmetric polynomials on $X$ is factorial. We consider conditions when a subsymmetric function on a Banach space can be approximated by subsymmetric analytic functions or polynomials. In addition we construct some weighted backward shift-like mappings on the metric space of point evaluation functionals on $\mathcal{P}_{\mathfrak{S}}(X)$ and prove their topological transitivity.