MULTIPLICATIVE FUNCTIONS ON HILBERT SPACES OF SYMMETRIC ANALYTIC FUNCTIONS

Abstract

We investigate multiplicative linear functionals on a Hilbert space of symmetric analytic functions on the Banach space of all absolutely summing sequences.  An Euclidian structure on the space of symmetric polynomials can be obtained if we suppose that all power polynomials are orthogonal. The completion of this space gives us a Hilbert space of symmetric analytic functions. A linear functional is multiplicative if it is multiplicative on the subspace of symmetric polynomials, which is considered algebra with respect to pointwise multiplication. That is, a linear multiplicative functional preserves the operations of addition and multiplication on the subspace of polynomials. It is shown that there is no multiplicative norm on the Hilbert space of symmetric analytic functions such that the radius of uniform convergence of all functions is infinite. The main goal of the paper is to establish necessary and sufficient conditions for the continuity of a linear multiplicative functional on the Hilbert space of symmetric analytic functions, providing that the Hilbert norm is multiplicative.  Also, we construct an example of a continuous multiplicative functional that is not an evaluation at any point of the space. Necessary and sufficient conditions for the continuity of linear multiplicative functionals on the Hilbert space of symmetric analytic functions are found. We compare multiplicative functionals of the Hilbert space of symmetric analytic functions with characters of the algebra of symmetric analytic functions of bounded type and find out that the sets of these functionals have nonempty intersection but do not contain each other. Finally, we consider, more precisely, multiplicative functionals that are not point evaluations. In particular, we show that any point evaluation functional can be represented as a linear combination of functionals of special form.