Analytic mappings on infinite-dimensional Banach spaces is an important part of contemporary functional analysis. The interest to actions of operator groups and semi-groups in spaces of analytic functions on an infinite-dimensional Banach space X increases during last years. The reasonable question here is about invariant subspaces of analytic functions on X and their algebraic and topological structures. In many cases such subspaces may have the structure of algebra with pointwise addition and multiplication and their spectra (sets of maximal ideals) has an important information about action of the given operator semi-group on X.
Algebras of analytic functions on Banach spaces and their spectra were invistigated by L. Nachbin, T. Gamelin, T. Corn, B. Cole, J. Mujica. Later, R. Aron, B. Cole and T, Gamelin considered the algebra Hb(X) of analytic functions of bounded type on a complex Banach space X and proposed to investigate the spectrum of a such algebra with using so-called the Aron-Berner extension of analytic functions of bounded type to the second dual space X** to X. This approach was used later by many authors. In particular, R. Aron, P. Galindo, D. Garcia and M. Maestre described the structures of analytic manifold over X** on the spectrum of Hb(X). This method was generalized in works of A.V. Zagorodnyuk and the Aron-Berner extension was used for polynomials on topological tensor products.
Symmetric polynomials on a Banach space were investigated by A. Nimerovski, S. Semenov, M. Gonzalez, R. Gonzalo, J, Jaramillo, R. Alencar, R. Aron, P. Galindo, A. Zagorodnyuk, M. Maestre, D. Garcia, P. Hajek, I. Chernega, T. Vasylyshyn, v. Kravtsiv and others. It is known that the spectrum of the algebra of symmetric analytic functions of bounded type on a Banach space is depending on the space and on the group or semi-group of symmetry. For example, P.Galindo, T. Vasylyshyn and A. Zagorodnyuk proved that if X = L∞[0,1], and the group of symmetry consists of measurable automorphisms of the interval [0,1], then the spectrum of the algebra of symmetric analytic functions of bounded type can be completely described by point evaluation functionals at points of X. On the other hand, I. Chernega, P. Galindo and A. Zagorodnyuk show that if X = lp, 1 ≤ p < ∞, and the group G is the group of all permutations of basis vectors, then the spectrum of the algebra of symmetric analytic functions of bounded type is larger than the set of point evaluation functionals. In this case, there are natural algebraic operation on the spectrum which form a unital commutative semi-ring structure.
