scholarly journals On the Bishop-Phelps-Bollobás Property for Numerical Radius

2014 ◽  
Vol 2014 ◽  
pp. 1-15 ◽  
Author(s):  
Sun Kwang Kim ◽  
Han Ju Lee ◽  
Miguel Martín
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Separable Banach Space ◽  
Sufficient Conditions ◽  
The Other ◽  
Numerical Radius ◽  
Infinite Dimensional ◽  
Other Hand ◽  
Nikodym Property

We study the Bishop-Phelps-Bollobás property for numerical radius (in short, BPBp-nu) and find sufficient conditions for Banach spaces to ensure the BPBp-nu. Among other results, we show thatL1μ-spaces have this property for every measureμ. On the other hand, we show that every infinite-dimensional separable Banach space can be renormed to fail the BPBp-nu. In particular, this shows that the Radon-Nikodým property (even reflexivity) is not enough to get BPBp-nu.

scholarly journals Classes of Entire Analytic Functions of Unbounded Type on Banach Spaces

Axioms ◽  
2020 ◽  
Vol 9 (4) ◽  
pp. 133
Author(s):  
Andriy Zagorodnyuk ◽  
Anna Hihliuk
Keyword(s):  
Banach Space ◽  
Analytic Function ◽  
Banach Spaces ◽  
Analytic Functions ◽  
Sufficient Conditions ◽  
Main Question ◽  
Dimensional Banach Space ◽  
Infinite Dimensional ◽  
Infinite Dimensional Banach Space ◽  
Taylor Polynomials

In this paper we investigate analytic functions of unbounded type on a complex infinite dimensional Banach space X. The main question is: under which conditions is there an analytic function of unbounded type on X such that its Taylor polynomials are in prescribed subspaces of polynomials? We obtain some sufficient conditions for a function f to be of unbounded type and show that there are various subalgebras of polynomials that support analytic functions of unbounded type. In particular, some examples of symmetric analytic functions of unbounded type are constructed.

scholarly journals AN OPERATOR SUMMABILITY OF SEQUENCES IN BANACH SPACES

2013 ◽  
Vol 56 (2) ◽  
pp. 427-437 ◽  
Author(s):  
ANIL KUMAR KARN ◽  
DEBA PRASAD SINHA
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Borel Measure ◽  
Operator Ideal ◽  
The Other ◽  
Summable Sequence ◽  
Other Hand ◽  
Limited Operator

AbstractLet 1 ≤ p < ∞. A sequence 〈 xn 〉 in a Banach space X is defined to be p-operator summable if for each 〈 fn 〉 ∈ lw*p(X*) we have 〈〈 fn(xk)〉k〉n ∈ lsp(lp). Every norm p-summable sequence in a Banach space is operator p-summable whereas in its turn every operator p-summable sequence is weakly p-summable. An operator T ∈ B(X, Y) is said to be p-limited if for every 〈 xn 〉 ∈ lpw(X), 〈 Txn 〉 is operator p-summable. The set of all p-limited operators forms a normed operator ideal. It is shown that every weakly p-summable sequence in X is operator p-summable if and only if every operator T ∈ B(X, lp) is p-absolutely summing. On the other hand, every operator p-summable sequence in X is norm p-summable if and only if every p-limited operator in B(lp', X) is absolutely p-summing. Moreover, this is the case if and only if X is a subspace of Lp(μ) for some Borel measure μ.

scholarly journals Nonwandering operators in Banach space

2005 ◽  
Vol 2005 (24) ◽  
pp. 3895-3908 ◽  
Author(s):  
Lixin Tian ◽  
Jiangbo Zhou ◽  
Xun Liu ◽  
Guangsheng Zhong
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Sequence Space ◽  
Separable Banach Space ◽  
Infinite Dimensional ◽  
Banach Sequence Space ◽  
Background Space ◽  
Physical Background ◽  
Structurally Stable ◽  
Banach Sequence

We introduce nonwandering operators in infinite-dimensional separable Banach space. They are new linear chaotic operators and are relative to hypercylic operators, but different from them. Firstly, we show some examples for nonwandering operators in some typical infinite-dimensional Banach spaces, including Banach sequence space and physical background space. Then we present some properties of nonwandering operators and the spectra decomposition of invertible nonwandering operators. Finally, we obtain that invertible nonwandering operators are locally structurally stable.

scholarly journals Strongly exposed points in bases for the positive cone of ordered Banach spaces and characterizations of l1(Г)

1986 ◽  
Vol 29 (2) ◽  
pp. 271-282 ◽  
Author(s):  
Ioannis A. Polyrakis
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Convex Sets ◽  
Extreme Points ◽  
Positive Cone ◽  
Infinite Dimensional ◽  
Strongly Exposed Points ◽  
Nikodym Property ◽  
Choquet Theory ◽  
Ordered Banach Spaces

The study of extreme, strongly exposed points of closed, convex and bounded sets in Banach spaces has been developed especially by the interconnection of the Radon–Nikodým property with the geometry of closed, convex and bounded subsets of Banach spaces [5],[2] . In the theory of ordered Banach spaces as well as in the Choquet theory, [4], we are interested in the study of a special type of convex sets, not necessarily bounded, namely the bases for the positive cone. In [7] the geometry (extreme points, dentability) of closed and convex subsets K of a Banach space X with the Radon-Nikodým property is studied and special emphasis has been given to the case where K is a base for acone P of X. In [6, Theorem 1], it is proved that an infinite-dimensional, separable, locally solid lattice Banach space is order-isomorphic to l1 if, and only if, X has the Krein–Milman property and its positive cone has a bounded base.

scholarly journals Exact Filling of Figures with the Derivatives of Smooth Mappings Between Banach Spaces

2005 ◽  
Vol 48 (4) ◽  
pp. 481-499
Author(s):  
D. Azagra ◽  
M. Fabian ◽  
M. Jiménez-Sevilla
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Sufficient Conditions ◽  
Smooth Mapping ◽  
Dimensional Banach Space ◽  
Infinite Dimensional ◽  
Smooth Case ◽  
Open Set ◽  
Infinite Dimensional Banach Space ◽  
Finite Dimensional Case

AbstractWe establish sufficient conditions on the shape of a set A included in the space (X, Y ) of the n-linear symmetric mappings between Banach spaces X and Y , to ensure the existence of a Cn-smooth mapping f: X → Y, with bounded support, and such that f(n)(X) = A, provided that X admits a Cn-smooth bump with bounded n-th derivative and dens X = dens ℒn(X, Y ). For instance, when X is infinite-dimensional, every bounded connected and open set U containing the origin is the range of the n-th derivative of such amapping. The same holds true for the closure of U, provided that every point in the boundary of U is the end point of a path within U. In the finite-dimensional case, more restrictive conditions are required. We also study the Fréchet smooth case for mappings from ℝn to a separable infinite-dimensional Banach space and the Gâteaux smooth case for mappings defined on a separable infinite-dimensional Banach space and with values in a separable Banach space.

scholarly journals On Basis Constants and Duality in Banach Spaces

1979 ◽  
Vol 22 (4) ◽  
pp. 459-465
Author(s):  
Leonard E. Dor
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Unconditional Basis ◽  
The Other ◽  
Other Hand ◽  
Basis Constant

AbstractEvery Banach space with a non-shrinking (unconditional) basis (Xi) can be renormed so that the biorthogonal sequence has a much smaller (unconditional) basis constant than (xi). On the other hand, if the unconditional constant of is C < 2 then the unconditional constant of (xi) is at most C/(2—C). This estimate is sharp.

Left approximate identities in algebras of compact operators on Banach spaces

1986 ◽  
Vol 104 (1-2) ◽  
pp. 169-175 ◽  
Author(s):  
P. G. Dixon
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Banach Spaces ◽  
Finite Rank ◽  
Sufficient Conditions ◽  
Compact Operators ◽  
The Other ◽  
Compact Set ◽  
Approximate Identities ◽  
Necessary And Sufficient

SynopsisWe study the existence of left approximate units, left approximate identities and bounded left approximate identities in the algebras (X)of all compact operators on a Banach space X and ℱ(X)− of all operators uniformly approximable by finite rank operators. In the case of bounded left approximate identities, necessary and sufficient conditions on X are obtained. In the other cases, sufficient conditions are obtained, together with an example of non-existence using a space constructed by Szankowski. The possibility of the sufficient conditions being also necessary depends on the question of whether every compact set is contained in the closure of the image of the unit ball under an operator in (X)(or ℱ(X)−). Sufficient conditions on X are obtained for this to be true, but it is conjectured that the answer for general X is negative.

Unconditional bases and martingales in LP(F)

1979 ◽  
Vol 85 (1) ◽  
pp. 117-123 ◽  
Author(s):  
D. J. Aldous
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Separable Banach Space ◽  
Unconditional Basis ◽  
Nice Property ◽  
Unconditional Bases ◽  
Measurable Functions ◽  
Nikodym Property

Here we describe our results and their background: terminology (mostly standard) is denned in Section 2. Throughout, F is a separable Banach space, 1 ≤ p < ∞ and Lp(F) is the space of measurable functions [0,1] → F with P-integrable norms. Given a ‘nice’ property P for Banach spaces, we may formulate the conjecture: Lp(F) satisfies P if and only if both F and Lp (= LP(ℝ)) satisfy P. This conjecture is known to be true for various specific properties, for example the Radon–Nikodym property ((4), section 5·4); reflexivity ((4), corollary 4·1·2); super-refiexivity ((12), proposition 1·2); B-convexity ((14), p. 200); and the properties of not containing copies of c0 (6) and l1 (13). The object of this paper is to demonstrate that the conjecture is false for the property of having an unconditional basis – this answers a question in (4).

Average Shadowing Property and Asymptotic Average Shadowing Property of Linear Dynamical Systems

2019 ◽  
Vol 29 (12) ◽  
pp. 1950170
Author(s):  
Lixin Jiao ◽  
Lidong Wang ◽  
Fengquan Li
Keyword(s):  
Banach Space ◽  
Dynamical Systems ◽  
Banach Spaces ◽  
Sufficient Conditions ◽  
Invertible Operator ◽  
Linear Dynamical Systems ◽  
Shadowing Property ◽  
Infinite Dimensional ◽  
Asymptotic Average Shadowing ◽  
Necessary And Sufficient

This paper investigates the average shadowing property and the asymptotic average shadowing property of linear dynamical systems in Banach spaces. Firstly, necessary and sufficient conditions for an invertible operator [Formula: see text] on a Banach space to have the average shadowing property and the asymptotic average shadowing property are given, respectively. Then, it is concluded that both the average shadowing property and the asymptotic average shadowing property are preserved under iterations. Furthermore, if [Formula: see text] is hyperbolic, then [Formula: see text] has the (asymptotic) average shadowing property. However, the inverse implication fails in infinite-dimensional Banach spaces. Finally, it is proved that the (asymptotic) average shadowing property is equivalent to the hyperbolicity for dynamical systems in a finite-dimensional Banach space.

scholarly journals On a New Geometric Constant Related to the Euler-Lagrange Type Identity in Banach Spaces

Mathematics ◽  
2021 ◽  
Vol 9 (2) ◽  
pp. 116
Author(s):  
Qi Liu ◽  
Yongjin Li
Keyword(s):  
Banach Spaces ◽  
Product Space ◽  
Sufficient Conditions ◽  
Normal Structure ◽  
The Other ◽  
Inner Product ◽  
Equivalent Characterization ◽  
Geometric Constant ◽  
Geometric Constants

In this paper, we will introduce a new geometric constant LYJ(λ,μ,X) based on an equivalent characterization of inner product space, which was proposed by Moslehian and Rassias. We first discuss some equivalent forms of the proposed constant. Next, a characterization of uniformly non-square is given. Moreover, some sufficient conditions which imply weak normal structure are presented. Finally, we obtain some relationship between the other well-known geometric constants and LYJ(λ,μ,X). Also, this new coefficient is computed for X being concrete space.

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