@article{Halushchak_2019, title={Spectra of some algebras of entire functions of bounded type, generated by a sequence of polynomials: Array}, volume={11}, url={https://journals.pnu.edu.ua/index.php/cmp/article/view/2110}, DOI={10.15330/cmp.11.2.311-320}, abstractNote={<p>In this work, we investigate the properties of the topological algebra of entire functions of bounded type, generated by a countable set of homogeneous polynomials on a complex Banach space.</p> <p>Let $X$ be a complex Banach space. We consider a subalgebra $H_{b\mathbb{P }(X)$ of the Fréchet algebra of entire functions of bounded type $H_b(X),$ generated by a countable set of algebraically independent homogeneous polynomials $\mathbb{P}.$ We show that each term of the Taylor series expansion of entire function, which belongs to the algebra $H_{b\mathbb{P }(X),$ is an algebraic combination of elements of $\mathbb{P}.$ We generalize the theorem for computing the radius function of a linear functional on the case of arbitrary subalgebra of the algebra $H_b(X)$ on the space $X.$ Every continuous linear multiplicative functional, acting from $H_{b\mathbb{P }(X)$ to $\mathbb{C}$ is uniquely determined by the sequence of its values on the elements of $\mathbb{P}.$ Consequently, there is a bijection between the spectrum (the set of all continuous linear multiplicative functionals) of the algebra $H_{b\mathbb{P }(X)$ and some set of sequences of complex numbers. We prove the upper estimate for sequences of this set. Also we show that every function that belongs to the algebra $H_{b\mathbb{P }(X),$ where $X$ is a closed subspace of the space $\ell_{\infty}$ such that $X$ contains the space $c_{00},$ can be uniquely analytically extended to $\ell_{\infty}$ and algebras $H_{b\mathbb{P }(X)$ and $H_{b\mathbb{P }(\ell)$ are isometrically isomorphic. We describe the spectrum of the algebra $H_{b\mathbb{P }(X)$ in this case for some special form of the set $\mathbb{P}.$</p> <p>Results of the paper can be used for investigations of the algebra of symmetric analytic functions on Banach spaces.</p>}, number={2}, journal={Carpathian Mathematical Publications}, author={Halushchak, S.I.}, year={2019}, month={Dec.}, pages={311–320} }