ON SOME COUNTABLY GENERATED ALGEBRA OF ENTIRE FUNCTIONS OF BOUNDED TYPE ON THE SPACE OF ALL BOUNDED SEQUENCES OF COMPLEX NUMBERS AND ITS SPECTRUM
DOI:
https://doi.org/10.31471/2304-7399-2025-21(79)-41-51Keywords:
n-homogeneous polynomial, symmetric polynomial, analytic function, spectrum of algebra.Abstract
Let $\ell_\infty$ be the complex Banach space of all bounded sequences of complex numbers. In this paper we study the Fr\'{e}chet subalgebra $H_{b\mathcal{P}}(\ell_\infty)$ of the Fr\'{e}chet algebra of all entire functions of bounded type $H_b(\ell_\infty),$ generated by some fixed countable set $\mathcal{P}$
of continuous algebraically independent complex-valued homogeneous polynomials on the space $\ell_\infty$ 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
$H_{b\mathcal{P}}(\ell_\infty)$ 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 $H_{b\mathcal{P}}(\ell_\infty).$
References
1. Aron R.M., Cole B.J., Gamelin T.W. Spectra of algebras of analytic functions on a Banach space, J. Reine Angew. Math., 415 (1991), 51 – 93.
2. Bandura A., Kravtsiv V., Vasylyshyn T. Algebraic basis of the algebra of all symmetric continuous polynomials on the Cartesian product of ℓp-spaces, Axioms, 11 (2022), no. 2, art. no. 41. doi:10.3390/axioms11020041
3. Chernega I., Galindo P., Zagorodnyuk A. On the spectrum of the algebra of bounded-type symmetric analytic functions on ℓ1, Math. Nachr., 297 (2024), 3835–3846. doi:10.1002/mana.202300415
4. Galindo P., Vasylyshyn T., Zagorodnyuk A. The algebra of symmetric analytic functions on L∞, Proc. Roy. Soc. Edinburgh Sect. A, 147 (2017), no. 4, 743–761. doi:10.1017/S0308210516000287
5. Galindo P., Vasylyshyn T., Zagorodnyuk A. Analytic structure on the spectrum of the algebra of symmetric analytic functions on L∞, Rev. R. Acad. Cienc. Exactas F´is. Nat. Ser. A Mat., 114 (2020), art. no. 56. doi:10.1007/s13398-020-00791-w
6. Gonz´alez M., Gonzalo R., Jaramillo J. A. Symmetric polynomials on rearrangement invariant function spaces, J. Lond. Math. Soc., 59 (1999), no. 2, 681–697. doi:10.1112/S0024610799007164
7. Halushchak S.I. Spectra of some algebras of entire functions of ounded type, generated by a sequence of polynomials, Carpathian Math. Publ., 11 (2019), no. 2, 311 – 320. doi:10.15330/cmp.11.2.311-320
8. Halushchak S.I. Isomorphisms of some algebras of analytic functions оf bounded type on Banach spaces, Mat. Stud., 56 (2021), no. 1, 107 – 12. doi:10.30970/ms.56.1.106-112
9. Kravtsiv V. V. Zeros of block-symmetric polynomials on Banach spaces, Mat. Stud., 53 (2020), no. 2, 206 – 211. doi:10.30970/ms.53.2.206-211
10. Kravtsiv V. Block-supersymmetric polynomials on spaces of absolutely convergent series, Symmetry, 16 (2024), no. 2, art. no. 179. doi:10.3390/sym16020179
11. Kravtsiv V., Vasylyshyn T., Zagorodnyuk A. On algebraic basis of the algebra of symmetric polynomials on ℓp(Cn), J. Funct. Spaces, 2017 (2017), Article ID 4947925, 8 p. doi:10.1155/2017/4947925
12. Kravtsiv V. V., Zagorodnyuk A. V. Spectra of algebras of blocksymmetric analytic functions of bounded type, Mat. Stud., 58 (2022), no. 1, 69–81. doi:10.30970/ms.58.1.69-81
13. Mujica J. Complex Analysis in Banach Spaces. North Holland, 1986.
14. Nemirovskii A.S., Semenov S.M. On polynomial approximation of functions on Hilbert space, Mat. USSR Sbornik, 21 (1973), no. 2, 255–277. doi:10.1070/SM1973v021n02ABEH002016
15. Novosad Z., Vasylyshyn S., Zagorodnyuk A. Countably generated algebras of analytic functions on Banach spaces, Axioms, 12 (2023), no. 8, art. no. 798. doi:10.3390/axioms12080798
16. Rudin W. Functional Analysis 2nd ed.. International series in pure and applied mathematics, 1991.
17. Stein E.M., Shakarchi R. Complex Analysis. Princeton University Press, 2003.
18. Vasylyshyn S.I. Spectra of algebras of analytic functions, generated by sequences of polynomials on Banach spaces and operations on spectra, Carpathian Math. Publ., 15 (2023), no. 1, 104 – 119. doi:10.15330/cmp.15.1.104-119
19. Vasylyshyn S.I. Topological isomorphism of a countably generated algebra of entire functions on ℓ∞ and the algebra of symmetric entire functions on L2∞ [0,1], Carpathian Math. Publ. (to appear)
20. Vasylyshyn T. V. Symmetric polynomials on the Cartesian power of Lp on the semi-axis, Mat. Stud., 50 (2018), no. 1, 93–104. doi:10.15330/ms.50.1.93-104
21. Vasylyshyn T. The algebra of symmetric polynomials on (L∞)n, Mat. Stud., 52 (2019), no. 1, 71 – 85. doi:10.30970/ms.52.1.71-85
22. Vasylyshyn T. Symmetric polynomials on (Lp)n, Eur. J. Math., 6 (2020), no. 1, 164–178. doi:10.1007/s40879-018-0268-3
23. Vasylyshyn T.V., Zagorodnyuk A.V. Symmetric polynomials on the Cartesian power of the real Banach space L∞[0,1], Mat. Stud., 53 (2020), no. 2, 192–205. doi:10.30970/ms.53.2.192-205
24. Vasylyshyn T. Symmetric analytic functions on the Cartesian power of the complex Banach space of Lebesgue measurable essentially bounded functions on [0,1], Journal of Mathematical Analysis and Applications, 509 (2022), no. 2, 125977. doi:10.1016/j.jmaa.2021.125977
25. Vasylyshyn T., Zahorodniuk V. On isomorphisms of algebras of entire symmetric functions on Banach spaces, J. Math. Anal. Appl., 529 (2024), no. 2, art. no. 127370. doi:10.1016/j.jmaa.2023.127370
26. Vasylyshyn T.V. Isomorphisms of algebras of symmetric functions on spaces ℓp, Mat. Stud., 63 (2025), no. 1, 77–87. doi:10.30970/ms.63.1.77-87
27. Zagorodnyuk A.V. Spectra of Algebras of Entire Functions on Banach Spaces, Proc. Amer. Math. Soc., 134 (2006), 2559 – 2569. doi:10.1090/S0002-9939-06-08260-8
28. Zagorodnyuk A.V., Kravtsiv V.V. Multiplicative Convolution on the Algebra of Block-Symmetric Analytic Functions, J. Math. Sci., 246 (2020), no. 2, 245 – 255. doi:10.1007/s10958-020-04734-z
