scholarly journals THE NORM OF THE PRODUCT OF POLYNOMIALS IN INFINITE DIMENSIONS

2006 ◽  
Vol 49 (1) ◽  
pp. 17-28 ◽  
Author(s):  
C. Boyd ◽  
R. A. Ryan
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Uniform Convexity ◽  
Analogous Problem ◽  
Homogeneous Polynomials ◽  
Infinite Dimensions ◽  
Multilinear Maps ◽  
Local Reflexivity ◽  
Positive Integers

AbstractGiven a Banach space $E$ and positive integers $k$ and $l$ we investigate the smallest constant $C$ that satisfies $\|P\|\hskip1pt\|Q\|\le C\|PQ\|$ for all $k$-homogeneous polynomials $P$ and $l$-homogeneous polynomials $Q$ on $E$. Our estimates are obtained using multilinear maps, the principle of local reflexivity and ideas from the geometry of Banach spaces (type and uniform convexity). We also examine the analogous problem for general polynomials on Banach spaces.

scholarly journals THE POLYNOMIAL NUMERICAL INDEX OF A BANACH SPACE

2006 ◽  
Vol 49 (1) ◽  
pp. 39-52 ◽  
Author(s):  
Yun Sung Choi ◽  
Domingo Garcia ◽  
Sung Guen Kim ◽  
Manuel Maestre
Keyword(s):  
Banach Space ◽  
Positive Integer ◽  
Banach Spaces ◽  
Compact Space ◽  
Linear Operators ◽  
Homogeneous Polynomials ◽  
Numerical Radius ◽  
Disc Algebra ◽  
Multilinear Maps ◽  
Numerical Index

AbstractIn this paper, we introduce the polynomial numerical index of order $k$ of a Banach space, generalizing to $k$-homogeneous polynomials the ‘classical’ numerical index defined by Lumer in the 1970s for linear operators. We also prove some results. Let $k$ be a positive integer. We then have the following:(i) $n^{(k)}(C(K))=1$ for every scattered compact space $K$.(ii) The inequality $n^{(k)}(E)\geq k^{k/(1-k)}$ for every complex Banach space $E$ and the constant $k^{k/(1-k)}$ is sharp.(iii) The inequalities$$ n^{(k)}(E)\leq n^{(k-1)}(E)\leq\frac{k^{(k+(1/(k-1)))}}{(k-1)^{k-1}}n^{(k)}(E) $$for every Banach space $E$.(iv) The relation between the polynomial numerical index of $c_0$, $l_1$, $l_{\infty}$ sums of Banach spaces and the infimum of the polynomial numerical indices of them.(v) The relation between the polynomial numerical index of the space $C(K,E)$ and the polynomial numerical index of $E$.(vi) The inequality $n^{(k)}(E^{**})\leq n^{(k)}(E)$ for every Banach space $E$.Finally, some results about the numerical radius of multilinear maps and homogeneous polynomials on $C(K)$ and the disc algebra are given.

scholarly journals Isomorphisms of some algebras of analytic functions of bounded type on Banach spaces

2021 ◽  
Vol 56 (1) ◽  
pp. 106-112
Author(s):  
S.I. Halushchak
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Analytic Functions ◽  
Entire Functions ◽  
Complex Banach Space ◽  
Homogeneous Polynomials ◽  
Topological Algebras ◽  
Nonlinear Functional

The theory of analytic functions is an important section of nonlinear functional analysis.In many modern investigations topological algebras of analytic functions and spectra of suchalgebras are studied. In this work we investigate the properties of the topological algebras of entire functions,generated by countable sets of homogeneous polynomials on complex Banach spaces. Let $X$ and $Y$ be complex Banach spaces. Let $\mathbb{A}= \{A_1, A_2, \ldots, A_n, \ldots\}$ and $\mathbb{P}=\{P_1, P_2,$ \ldots, $P_n, \ldots \}$ be sequences of continuous algebraically independent homogeneous polynomials on spaces $X$ and $Y$, respectively, such that $\|A_n\|_1=\|P_n\|_1=1$ and $\deg A_n=\deg P_n=n,$ $n\in \mathbb{N}.$ We consider the subalgebras $H_{b\mathbb{A}}(X)$ and $H_{b\mathbb{P}}(Y)$ of the Fr\'{e}chet algebras $H_b(X)$ and $H_b(Y)$ of entire functions of bounded type, generated by the sets $\mathbb{A}$ and $\mathbb{P}$, respectively. It is easy to see that $H_{b\mathbb{A}}(X)$ and $H_{b\mathbb{P}}(Y)$ are the Fr\'{e}chet algebras as well. In this paper we investigate conditions of isomorphism of the topological algebras $H_{b\mathbb{A}}(X)$ and $H_{b\mathbb{P}}(Y).$ We also present some applications for algebras of symmetric analytic functions of bounded type. In particular, we consider the subalgebra $H_{bs}(L_{\infty})$ of entire functions of bounded type on $L_{\infty}[0,1]$ which are symmetric, i.e. invariant with respect to measurable bijections of $[0,1]$ that preserve the measure. We prove that$H_{bs}(L_{\infty})$ is isomorphic to the algebra of all entire functions of bounded type, generated by countable set of homogeneous polynomials on complex Banach space $\ell_{\infty}.$

Complex Uniform Convexity and Riesz Measures

2004 ◽  
Vol 56 (2) ◽  
pp. 225-245 ◽  
Author(s):  
Gordon Blower ◽  
Thomas Ransford
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Mass Distribution ◽  
Complex Plane ◽  
Subharmonic Function ◽  
Uniform Convexity ◽  
Von Neumann ◽  
Riesz Measure

AbstractThe norm on a Banach space gives rise to a subharmonic function on the complex plane for which the distributional Laplacian gives a Riesz measure. This measure is calculated explicitly here for LebesgueLpspaces and the von Neumann-Schatten trace ideals. Banach spaces that areq-uniformly PL-convex in the sense of Davis, Garling and Tomczak-Jaegermann are characterized in terms of the mass distribution of this measure. This gives a new proof that the trace idealscpare 2-uniformly PL-convex for 1 ≤p≤ 2.

scholarly journals Orbits of homogeneous polynomials on Banach spaces

2020 ◽  
pp. 1-29
Author(s):  
RODRIGO CARDECCIA ◽  
SANTIAGO MURO
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Homogeneous Polynomial ◽  
Julia Sets ◽  
Accumulation Point ◽  
Dense Orbit ◽  
Homogeneous Polynomials ◽  
Infinite Dimensional

We study the dynamics induced by homogeneous polynomials on Banach spaces. It is known that no homogeneous polynomial defined on a Banach space can have a dense orbit. We show a simple and natural example of a homogeneous polynomial with an orbit that is at the same time $\unicode[STIX]{x1D6FF}$ -dense (the orbit meets every ball of radius $\unicode[STIX]{x1D6FF}$ ), weakly dense and such that $\unicode[STIX]{x1D6E4}\cdot \text{Orb}_{P}(x)$ is dense for every $\unicode[STIX]{x1D6E4}\subset \mathbb{C}$ that either is unbounded or has 0 as an accumulation point. Moreover, we generalize the construction to arbitrary infinite-dimensional separable Banach spaces. To prove this, we study Julia sets of homogeneous polynomials on Banach spaces.

scholarly journals The fixed point property in banach spaces whose characteristic of uniform convexity is less than 2

1993 ◽  
Vol 54 (2) ◽  
pp. 169-173 ◽  
Author(s):  
J. García Falset
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Banach Spaces ◽  
Orthogonality Condition ◽  
Fixed Point Property ◽  
Uniform Convexity ◽  
Point Property

AbstractWe prove that every Banach space X with characteristic of uniform convexity less than 2 has the fixed point property whenever X satisfies a certain orthogonality condition.

Bounds on the derivatives of polynomials on Banach spaces

1991 ◽  
Vol 110 (2) ◽  
pp. 307-312 ◽  
Author(s):  
Yannis Sarantopoulos
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Real Banach Space ◽  
Homogeneous Polynomials ◽  
Derivatives Of

AbstractWe generalize the classical Bernstein's and Markov's Inequalities for polynomials on any real Banach space. We also give estimates for the derivatives of homogeneous polynomials on real Banach spaces.

On martingales with values in a complex Banach space

1988 ◽  
Vol 104 (2) ◽  
pp. 399-406 ◽  
Author(s):  
D. J. H. Garling
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Complex Banach Space ◽  
Uniform Convexity ◽  
Martingale Inequalities ◽  
Nikodym Property

In recent years it has become clear that there are several ways in which complex Banach spaces can differ quite markedly from their real counterparts, and many of these concern martingales. Thus, in [6] complex uniform convexity was related to martingale inequalities, in [3] and [7] the convergence of L1-bounded analytic martingales was considered and in [8] this property was related to the analytic Radon–Nikodym property.

scholarly journals Nonlinear Weakly Sequentially Continuous Embeddings Between Banach Spaces

2018 ◽  
Vol 2020 (18) ◽  
pp. 5506-5533 ◽  
Author(s):  
B M Braga
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Uniform Convexity ◽  
Null Sequence ◽  
Uniform Embedding ◽  
Spreading Model ◽  
Continuous Map ◽  
Weakly Sequentially Continuous

Abstract In these notes, we study nonlinear embeddings between Banach spaces that are also weakly sequentially continuous. In particular, our main result implies that if a Banach space $X$ coarsely (resp. uniformly) embeds into a Banach space $Y$ by a weakly sequentially continuous map, then every spreading model $(e_n)_n$ of a normalized weakly null sequence in $X$ satisfies $$ \|e_1+\ldots+e_k\|_{\overline{\delta}_Y}\lesssim\|e_1+\ldots+e_k\|_S,$$where $\overline{\delta }_Y$ is the modulus of asymptotic uniform convexity of $Y$. Among other results, we obtain Banach spaces $X$ and $Y$ so that $X$ coarsely (resp. uniformly) embeds into $Y$, but so that $X$ cannot be mapped into $Y$ by a weakly sequentially continuous coarse (resp. uniform) embedding.

scholarly journals p-Uniform Convexity andq-Uniform Smoothness of Absolute Normalized Norms onℂ2

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Tomonari Suzuki
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Convex Banach Space ◽  
Uniform Convexity ◽  
Uniformly Convex Banach Space ◽  
Uniformly Convex ◽  
Uniform Smoothness

We first prove characterizations ofp-uniform convexity andq-uniform smoothness. We next give a formulation on absolute normalized norms onℂ2. Using these, we present some examples of Banach spaces. One of them is a uniformly convex Banach space which is notp-uniformly convex.

scholarly journals Note on bases in algebras of analytic functions on Banach spaces

2019 ◽  
Vol 11 (1) ◽  
pp. 42-47 ◽  
Author(s):  
I.V. Chernega ◽  
A.V. Zagorodnyuk
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Analytic Functions ◽  
Entire Functions ◽  
Complex Banach Space ◽  
Schauder Basis ◽  
Symmetric Polynomials ◽  
Homogeneous Polynomials ◽  
Absolute Basis

Let $\{P_n\}_{n=0}^\infty$ be a sequenceof continuous algebraically independent  homogeneous polynomials on a complex Banach space $X.$ We consider the following question: Under which conditions polynomials $\{P_1^{k_1}\cdots P_n^{k_n}\}$ form a Schauder (perhaps absolute) basis in the minimal subalgebra of entire functions of bounded type on $X$ which contains the sequence $\{P_n\}_{n=0}^\infty$? In the paper we study the following examples: when $P_n$ are coordinate functionals on $c_0,$ and when $P_n$ are symmetric polynomials on $\ell_1$ and on $L_\infty[0,1].$ We can see that for some cases $\{P_1^{k_1}\cdots P_n^{k_n}\}$ is a Schauder basis which is not absolute but for some cases it is absolute.

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