Block-symmetric and weakly symmetric polynomials on Banach spaces of absolutely summable in a power $p\in [1,+\infty)$ sequences
Abstract
The work is devoted to the study of block-symmetric and weakly symmetric continuous polynomials on the Banach space $\ell_p^{ (\mathbb{K}) }$ of all sequences $x = (x_1, x_2, \ldots)$ with $x_j \in \mathbb{K}$ for $j\in\mathbb{N}$ such that the series $\sum_{j=1}^\infty |x_j|^p$ is convergent, where $\mathbb{K} \in \{\mathbb{R}, \mathbb{C}\}$ and $p\in [1,+\infty).$
We construct an algebraic basis of the algebra of $n$-block-symmetric continuous polynomials on $\ell_p^{ (\mathbb{R}) }.$ Also we construct a generating system of the algebra of weakly symmetric continuous polynomials on $\ell_p^{ (\mathbb{K}) }$ for $\mathbb{K} \in \{\mathbb{R}, \mathbb{C}\}$ and we show that elements of this system are linearly dependent.
We show that the weakly symmetric continuous linear functionals, in contrast to the block-symmetric functions, separate points of the space $\ell_1^{ (\mathbb{K}) },$ where $\mathbb{K} \in \{\mathbb{R}, \mathbb{C}\}.$
References
R. Aron, P. Galindo, D. Pinasco, I. Zalduendo, Group-Symmetric Holomorphic Functions on a Banach Space, Bull. Lond. Math. Soc. 48 (5) (2016), 779–796. https://doi.org/10.1112/blms/bdw043
I.V. Burtnyak, Yu.Yu. Chopyuk, S.I. Vasylyshyn, T.V. Vasylyshyn, Algebras of Weakly Symmetric Functions on Spaces of Lebesgue Measurable Functions, Carpathian Math. Publ. 15 (2) (2023), 411–419. https://doi.org/10.15330/cmp.15.2.411-419
I. Chernega, P. Galindo, A. Zagorodnyuk, On the Spectrum of the Algebra of Bounded-Type Symmetric Analytic Functions on $ell_1$, Math. Nachr. 297 (10) (2024), 3835–3846. https://doi.org/10.1002/mana.202300415
I. Chernega, M. Martsinkiv, T. Vasylyshyn, A. Zagorodnyuk, Applications of Supersymmetric Polynomials in Statistical Quantum Physics, Quantum Reports 5 (4) (2023), 683–697.
https://doi.org/10.3390/quantum5040043
D.Yu. Dolishniak, V.V. Kravtsiv, Subsymmetric Functions on Banach Spaces with Subsymmetric Bases, Carpathian Math. Publ. 18 (1) (2026), 49–66. https://doi.org/10.15330/cmp.18.1.49-66
R. Hryniv, V. Kravtsiv, T. Vasylyshyn, A. Zagorodnyuk, Symmetric and Supersymmetric Polynomials on $ell_p$ and Partition Functions in Quantum Statistical Physics, Physica Scripta 100 (7) (2025), Paper No. 075208. https://doi.org/10.1088/1402-4896/adde1e
V.V. Kravtsiv, A.V. Zagorodnyuk, Multiplicative Convolution on the Algebra of Block-Symmetric Analytic Functions, J. Math. Sci. (N.Y.) 246 (2) (2020), 245–255. https://doi.org/10.1007/s10958-020-04734-z
V.V. Kravtsiv, A.V. Zagorodnyuk, Spectra of Algebras of Block-Symmetric Analytic Functions of Bounded Type, Mat. Stud. 58 (1) (2022), 69–81. https://doi.org/10.30970/ms.58.1.69-81
S. Lang, Complex Analysis, 4th ed., Graduate Texts in Mathematics, V.103, Springer, New York, 1999.
R.V. Ponomarov, T.V. Vasylyshyn, Algebraic Bases of Some Algebras of Polynomials on Banach Spaces, Mat. Stud. 64 (1) (2025), 81–91. https://doi.org/10.30970/ms.64.1.81-91
J.J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, Cambridge, 1999.
S.I. Vasylyshyn, Topological Isomorphism of a Countably Generated Algebra of Entire Functions on $ell_infty$ and the Algebra of Symmetric Entire Functions on $L_infty^2[0,1]$, Carpathian Math. Publ. 18 (1) (2026), 78–98. https://doi.org/10.15330/cmp.18.1.78-98
T.V. Vasylyshyn, Isomorphisms of Algebras of Symmetric Functions on Spaces $ell_p$, Mat. Stud. 63 (1) (2025), 77–87. https://doi.org/10.30970/ms.63.1.77-87
T. Vasylyshyn, V. Zahorodniuk, Weakly Symmetric Functions on Spaces of Lebesgue Integrable Functions, Carpathian Math. Publ. 14 (2) (2022), 437–441. https://doi.org/10.15330/cmp.14.2.437-441
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