ON SOME COUNTABLY GENERATED ALGEBRA OF ENTIRE FUNCTIONS OF BOUNDED TYPE ON THE SPACE OF ALL BOUNDED SEQUENCES OF COMPLEX NUMBERS AND ITS SPECTRUM - Vasylyshyn, S. І. (PRECARPATHIAN BULLETIN OF THE SHEVCHENKO SCIENTIFIC SOCIETY. Number, 2025)
Let $\ell_∞$ be the complex Banach space of all bounded sequences of complex numbers. In this paper we study the Fréchet subalgebra $H_{b\mathcal{P}}(\ell_∞)$ of the Fréchet algebra of all entire functions of bounded type $H_b(\ell_∞),$ generated by some fixed countable set $\mathcal{P}$ of continuous algebraically independent complex-valued homogeneous polynomials on the space $\ell_∞$ such that the set $\mathcal{P}$ contains an infinite number of elements that share the same degree of homogeneity. We investigate the form of elements of this subalgebra. Furthermore, we show that every linear multiplicative functional, acting from $H_{b\mathcal{P}}(\ell_∞)$ to $\mathbb{C},$ can be uniquely identified with the sequence of its values on the elements of $\mathcal{P}$ and we prove that this sequence grows no faster than some geometric progression. We use these results to describe the spectrum of the algebra $H_{b\mathcal{P}}(\ell_∞).$