Some algebras of symmetric analytic functions and their spectra

Abstract

In the spectrum of the algebra of symmetric analytic functions of bounded type on p, 1 p < +∞, and along the same lines as the general non-symmetric case, we define and study a convolution operation and give a formula for the 'radius' function. It is also proved that the algebra of analytic functions of bounded type on 1 is isometrically isomorphic to an algebra of symmetric analytic functions on a polydisc of 1 . We also consider the existence of algebraic projections between algebras of symmetric polynomials and the corresponding subspace of subsymmetric polynomials.

Symmetric Polynomials and Holomorphic Functions on Infinite Dimensional Spaces

“…[6] Let us consider a sequence of real numbers (a n ), 0 ≤ |a n | < 1 such that (a n ) ∈ 2 \ 1 and that the series ∞ ∑ n=1 a n conditionally converges to some number C. Despite (a n ) / ∈ 1 , evaluations on (a n ) are determined for every symmetric polynomial on 1 . In particular,…”

“…In [6] we have introduced the analogous convolution in our symmetric setting. It is easy to see that if f is a symmetric function on p , then, in general, f (· + y) is not symmetric for a fixed y.…”

On Algebraic Basis of the Algebra of Symmetric Polynomials on lp(Cn)

“…Algebras of polynomials and analytic functions on a Banach space which are invariant (symmetric) with respect to a group of linear operators ( ) acting on were studied by a number of authors [1][2][3][4][5][6][7][8][9][10] (see also a survey [11]). If has a symmetric structure, then it is natural to consider the case when ( ) is a group of operators which preserve this structure.…”

“…Algebraic basis plays a crucial role in the problem of description of spectra of algebras generated by polynomials [1][2][3][4]. For example, each complex homomorphism on the algebra of symmetric polynomials on ℓ is completely defined by its values on the basis elements.…”

Homomorphisms of the algebra of symmetric analytic functions on $\ell_1$

“…Recall that for any ϕ, θ ∈ M bs (ℓ p ) and f ∈ H bs (ℓ p ), the symmetric convolution ϕ ⋆ θ was defined in [4] as follows c Chernega I., 2014 where…”