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 on  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, considered as an algebra with respect to the pointwise multiplication.  The main goal of the paper is to establish necessary and sufficient conditions for the continuity of a linear multiplicative functional on 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 . We compare multiplicative functionals of the Hilbert space of symmetric analytic functions with characters of the algebra of symmetric analytic functions of bounded type on  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 .