Symmetric analytic functions on Cartesian powers of complex Banach spaces of complex-valued Lebesgue integrable in a power p ∈ [0, + ∞) functions on [0, 1] and [0, + ∞)

Abstract

This work is devoted to the study of symmetric complex-valued analytic functions on Cartesian powers of complex Banach spaces Lp[0, 1] and Lp[0, + ∞) of all complex-valued Lebesgue integrable in a power p functions on [0, 1] and [0, + ∞) resp., where 1≤p< + ∞. We construct a finite algebraic basis of the algebra of all complex-valued symmetric continuous polynomials on the Cartesian power of the space Lp[0, 1] and we show that the Fréchet algebra of all complex-valued symmetric entire analytic functions on the Cartesian power of the space Lp[0, 1] is isomorphic to the Fréchet algebra H(Cm) of all complex-valued analytic functions on Cm, where m is the cardinality of the above-mentioned algebraic basis. Also we show that in the case p < N every complex-valued symmetric continuous polynomial on the Cartesian power of the space Lp[0, + ∞) is necessarily constant. In the case p ∈ N we establish results for symmetric continuous polynomials and symmetric analytic functions on the Cartesian power of the space Lp[0, + ∞), analogical to the results for symmetric continuous polynomials and symmetric analytic functions on the Cartesian power of the space Lp[0, 1].