scholarly journals Nonuniform dichotomy of evolutionary processes in Banach spaces

1983 ◽  
Vol 27 (1) ◽  
pp. 31-52 ◽  
Author(s):  
Petre Preda ◽  
Mihail Megan
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Evolutionary Processes ◽  
General Banach Space

In this paper we study nonuniform dichotomy concepts of linear evolutionary processes which are defined in a general Banach space and whose norms can increase no faster than an exponential. Connections between the dichotomy concepts and (B, D) admissibility properties are established. These connections have been partially accomplished in an earlier paper by the authors for the case when the process was a semigroup of class C0 and (B, D) = [(Lp, Lq).

scholarly journals Characterizations of metric projections in Banach spaces and applications

1998 ◽  
Vol 3 (1-2) ◽  
pp. 85-103 ◽  
Author(s):  
Jean-Paul Penot ◽  
Robert Ratsimahalo
Keyword(s):  
Banach Space ◽  
Variational Inequalities ◽  
Banach Spaces ◽  
Convex Subset ◽  
Metric Projection ◽  
Nonempty Closed Convex Subset ◽  
Convex Cones ◽  
Duality Mappings ◽  
Metric Projections ◽  
General Banach Space

This paper is devoted to the study of the metric projection onto a nonempty closed convex subset of a general Banach space. Thanks to a systematic use of semi-inner products and duality mappings, characterizations of the metric projection are given. Applications to decompositions of Banach spaces along convex cones and variational inequalities are presented.

On l1 subspaces of Orlicz vector-valued function spaces

1987 ◽  
Vol 101 (1) ◽  
pp. 107-112 ◽  
Author(s):  
Fernando Bombal
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
Function Spaces ◽  
Complete Characterization ◽  
Complemented Subspace ◽  
Vector Valued Function ◽  
Vector Valued ◽  
General Banach Space

The purpose of this paper is to characterize the Orlicz vector-valued function spaces containing a copy or a complemented copy of l1. Pisier proved in [13] that if a Banach space E contains no copy of l1, then the space Lp(S, Σ, μ, E) does not contain it either, for 1 < p < ∞. We extend this result to the case of Orlicz vector valued function spaces, by reducing the problem to the situation considered by Pisier. Next, we pass to study the problem of embedding l1 as a complemented subspace of LΦ(E). We obtain a complete characterization when E is a Banach lattice and only partial results in case of a general Banach space. We use here in a crucial way a result of E. Saab and P. Saab concerning the embedding of l1 as a complemented subspace of C(K, E), the Banach space of all the E-valued continuous functions on the compact Hausdorff space K (see [14]). Finally, we use these results to characterize several classes of Banach spaces for which LΦ(E) has some Banach space properties, namely the reciprocal Dunford-Pettis property and Pelczyński's V property.

Absolute and Unconditional Convergence in Normed Linear Spaces

1962 ◽  
Vol 58 (4) ◽  
pp. 575-579 ◽  
Author(s):  
D. Rutovitz
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Normed Linear Space ◽  
Unconditional Convergence ◽  
Geometrical Approach ◽  
Linear Spaces ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Open Question ◽  
General Banach Space

In 1933 Orlicz proved various results concerning unconditional convergence in Banach spaces (4), which were noted by Banach ((l), p. 240) who remarked that absolute and unconditional convergence are equivalent in finite dimensional Banach spaces, but that whether or not the two are non-equivalent in all infinite dimensional spaces was still an open question. MacPhail (3) gave a criterion for the equivalence of the two notions of convergence in a general Banach space and used it to prove non-equivalence in the spaces l1 and L1. In 1950 Dvoretzky and Rogers demonstrated the non-equivalence of the two types of convergence in any infinite dimensional normed linear space, using an elegant and instructive geometrical approach (2). The result has also been proved by a different method by Grothendieck (5).

Banach spaces of absolutely k-summable series

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Mehmet Ali Sarıgöl ◽  
Ravi P. Agarwal
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Hausdorff Measure ◽  
Measure Of Noncompactness ◽  
Compact Operators ◽  
Matrix Operator ◽  
Hausdorff Measure Of Noncompactness ◽  
Bk Space ◽  
New Space ◽  
General Banach Space

Abstract In this paper, we present a general Banach space of absolutely k-summable series using a triangle matrix operator and prove that this is a BK-space isometrically isomorphic to the space ℓ k {\ell_{k}} . We also establish the α - {\alpha-} , β - {\beta-} , γ-duals and base of the new space. Finally, we qualify some matrix and compact operators on the new space making use of the Hausdorff measure of noncompactness. Our results include, as particular cases, a number of well-known results.

scholarly journals Entire Analytic Functions of Unbounded Type on Banach Spaces and Their Lineability

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 150
Author(s):  
Andriy Zagorodnyuk ◽  
Anna Hihliuk
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Analytic Functions ◽  
Entire Functions ◽  
Dimensional Algebra ◽  
Dimensional Banach Space ◽  
Infinite Dimensional ◽  
Linear Subspaces ◽  
Infinite Dimensional Banach Space

In the paper we establish some conditions under which a given sequence of polynomials on a Banach space X supports entire functions of unbounded type, and construct some counter examples. We show that if X is an infinite dimensional Banach space, then the set of entire functions of unbounded type can be represented as a union of infinite dimensional linear subspaces (without the origin). Moreover, we show that for some cases, the set of entire functions of unbounded type generated by a given sequence of polynomials contains an infinite dimensional algebra (without the origin). Some applications for symmetric analytic functions on Banach spaces are obtained.

scholarly journals The dual of the space of bounded operators on a Banach space

2021 ◽  
Vol 8 (1) ◽  
pp. 48-59
Author(s):  
Fernanda Botelho ◽  
Richard J. Fleming
Keyword(s):  
Banach Space ◽  
Tensor Product ◽  
Banach Spaces ◽  
Dual Space ◽  
Tensor Products ◽  
Bounded Operators ◽  
Projective Tensor Product ◽  
Projective Tensor

Abstract Given Banach spaces X and Y, we ask about the dual space of the 𝒧(X, Y). This paper surveys results on tensor products of Banach spaces with the main objective of describing the dual of spaces of bounded operators. In several cases and under a variety of assumptions on X and Y, the answer can best be given as the projective tensor product of X ** and Y *.

The wigner property for CL-spaces and finite-dimensional polyhedral banach spaces

2021 ◽  
pp. 1-17
Author(s):  
Dongni Tan ◽  
Xujian Huang
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Phase Function ◽  
Linear Isometry ◽  
Basic Properties ◽  
Finite Dimensional ◽  
Image Position

Abstract We say that a map $f$ from a Banach space $X$ to another Banach space $Y$ is a phase-isometry if the equality \[ \{\|f(x)+f(y)\|, \|f(x)-f(y)\|\}=\{\|x+y\|, \|x-y\|\} \] holds for all $x,\,y\in X$ . A Banach space $X$ is said to have the Wigner property if for any Banach space $Y$ and every surjective phase-isometry $f : X\rightarrow Y$ , there exists a phase function $\varepsilon : X \rightarrow \{-1,\,1\}$ such that $\varepsilon \cdot f$ is a linear isometry. We present some basic properties of phase-isometries between two real Banach spaces. These enable us to show that all finite-dimensional polyhedral Banach spaces and CL-spaces possess the Wigner property.

Projections on Tree-Like banach Spaces

1985 ◽  
Vol 37 (5) ◽  
pp. 908-920
Author(s):  
A. D. Andrew
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Banach Spaces ◽  
Bounded Linear Operator ◽  
Unconditional Basis ◽  
Reflexive Banach Space ◽  
Separable Space ◽  
Bounded Linear ◽  
Tree Space

1. In this paper, we investigate the ranges of projections on certain Banach spaces of functions defined on a diadic tree. The notion of a “tree-like” Banach space is due to James 4], who used it to construct the separable space JT which has nonseparable dual and yet does not contain l1. This idea has proved useful. In [3], Hagler constructed a hereditarily c0 tree space, HT, and Schechtman [6] constructed, for each 1 ≦ p ≦ ∞, a reflexive Banach space, STp with a 1-unconditional basis which does not contain lp yet is uniformly isomorphic to for each n.In [1] we showed that if U is a bounded linear operator on JT, then there exists a subspace W ⊂ JT, isomorphic to JT such that either U or (1 — U) acts as an isomorphism on W and UW or (1 — U)W is complemented in JT. In this paper, we establish this result for the Hagler and Schechtman tree spaces.

scholarly journals ASYMPTOTIC PROPERTIES OF KOLMOGOROV WIDTHS

2010 ◽  
Vol 82 (1) ◽  
pp. 10-17
Author(s):  
MIKHAIL I. OSTROVSKII
Keyword(s):  
Banach Space ◽  
Asymptotic Behavior ◽  
Banach Spaces ◽  
Asymptotic Properties ◽  
Codimension One ◽  
Kolmogorov Widths

AbstractWe consider two problems concerning Kolmogorov widths of compacts in Banach spaces. The first problem is devoted to relations between the asymptotic behavior of the sequence of n-widths of a compact and of its projections onto a subspace of codimension one. The second problem is devoted to comparison of the sequence of n-widths of a compact in a Banach space 𝒴 and of the sequence of n-widths of the same compact in other Banach spaces containing 𝒴 as a subspace.

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.

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