Some properties of shift operators on algebras generated by ∗ -polynomials
Journal Title: Карпатські математичні публікації - Year 2018, Vol 10, Issue 1
Abstract
A ∗-polynomial is a function on a complex Banach space X, which is a sum of so-called (p,q)-polynomials.In turn, for non-negative integers p and q, a (p,q)-polynomial is a function on X, which is the restrictionto the diagonal of some mapping, defined on the Cartesian power Xp+q, which is linear with respect to every of its first p arguments and antilinear with respect to every of its other q arguments. The set of all continuous∗-polynomials on X form an algebra, which contains the algebra of all continuous polynomials on X as a propersubalgebra. So, completions of this algebra with respect to some natural norms are wider classes of functions thanalgebras of holomorphic functions. On the other hand, due to the similarity of structures of ∗-polynomials and polynomials, for the investigation of such completions one can use the technique, developed for the investigation of holomorphic functions on Banach spaces. We investigate the Fr\'{e}chet algebra of functions on a complex Banach space, which is the completion of the algebra of all continuous ∗-polynomials with respect to the countable system of norms, equivalent to norms of the uniform convergence on closed balls of the space. We establish some properties of shift operators (which act as the addition of some fixed element of the underlying space to the argument of a function) on this algebra. In particular, we show that shift operators are well-defined continuous linear operators. Also we prove some estimates for norms of values of shift operators. Using these results, we investigate one special class of functions from the algebra, which is important in the description of the spectrum (the set of all maximal ideals) of the algebra.
Authors and Affiliations
T. V. Vasylyshyn
Keywords
Related Articles
- EP ID EP532897
- DOI 10.15330/cmp.10.1.206-212
- Views 55
- Downloads 0
Mixed problem for the singular partial differential equation of parabolic type
The scheme for solving of a mixed problem is proposed for a differential equation a(x)∂T∂τ=∂∂x(c(x)∂T∂x)−g(x)T with coefficients a(x), g(x) that are the generalized derivatives of functions of bounded variation, c(x)>0,...
$\omega$-Euclidean domain and Laurent series
It is proved that a commutative domain $R$ is $\omega$-Euclidean if and only if the ring of formal Laurent series over $R$ is $\omega$-Euclidean domain. It is also proved that every singular matrice over ring of formal L...
Metric on the spectrum of the algebra of entire symmetric functions of bounded type on the complex $L_\infty$
It is known that every complex-valued homomorphism of the Fr\'{e}chet algebra $H_{bs}(L_\infty)$ of all entire symmetric functions of bounded type on the complex Banach space $L_\infty$ is a point-evaluation functional...
Uniform boundary controllability of a discrete 1-D Schrödinger equation
In this paper we study the controllability of a finite dimensional system obtained by discretizing in space and time the linear 1-D Schrodinger equation with a boundary control. As for other problems, we can expect that...
On a necessary condition for Lp (0<p<1) -convergence (upper boundedness) of trigonometric series
In this paper we prove that the condition ∑2nk=[n2]λk(p)(|n−k|+1)2−p=o(1)(=O(1)), is a necessary condition for the Lp(0<p<1)-convergence (upper boundedness) of a trigonometric series. Precisely, the results extend some r...