Spectra of some algebras of entire functions of bounded type, generated by a sequence of polynomials


  • S.I. Halushchak Vasyl Stefanyk Precarpathian National University, 57 Shevchenka str., 76018, Ivano-Frankivsk, Ukraine


$n$-homogeneous polynomial, analytic function, spectrum of algebra
Published online: 2019-12-31


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.

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 $\mathbb{P}.$

Results of the paper can be used for investigations of the algebra of symmetric analytic functions on Banach spaces.

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How to Cite
Halushchak, S. Spectra of Some Algebras of Entire Functions of Bounded Type, Generated by a Sequence of Polynomials. Carpathian Math. Publ. 2019, 11, 311-320.