On Algebraic Bases of Algebras of Block-symmetric
Year: 2012
Type: article
Abstract: In resent years there is increasing interest to investigations of invariants of permutation group S∞ of integer numbers. This group can be represented on a Banach space X with a symmetric basis as a group of operators of perturbation of basis vectors. The action of this group has a natural extension to the action on the algebra Hb(X) of analytic functions of bounded type on X. Invariants of this representation of S∞ are so called symmetric analytic functions of bounded type on X. The algebra of symmetric analytic functions Hbs(X) were investigated by many authors ([3], [4], [6]). In particular, it is known that Hbs(`p) admits an algebraic basis for 1 ≤ p <∞. On the other hand, there are more representations of S∞ in Banach spaces. For example, if X is a direct sum of infinitely many “blocks” which are copies of a Banach space X, then S∞ acts permutating the “blocks” (see for the definition below). For this case we have invariants — the algebra of block-symmetric analytic functions. Note that this algebra is much more complicated and in the general case has no algebraic basis (see e. g. [1, 2]). Note that if dimX <∞, then block-invariant polynomials are investigated in the classical theory of invariants [5, 7]. Let X = (∑ X )
Source: Matematychni Studii
Authors: Viktoriia Kravtsiv, Andriy Zagorodnyuk
Language: English
Topic: Advanced Banach Space Theory
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: diamond
OpenAlex ID: W201705186