Normal structure coefficients of Lp(Ω)

1991 ◽  
Vol 117 (3-4) ◽  
pp. 299-303 ◽  
Author(s):  
T. Domínguez Benavides
Keyword(s):  
Banach Space ◽  
Measure Space ◽  
Finite Measure ◽  
Normal Structure ◽  
Convex Banach Space ◽  
Uniformly Convex Banach Space ◽  
Uniformly Convex ◽  
Structure Coefficient ◽  
Finite Measure Space

SynopsisLet X be a uniformly convex Banach space, and N(X) the normal structure coefficient of X. In this paper it is proved that N(X) can be calculated by considering only sets whose points are equidistant from their Chebyshev centre. This result is applied to prove that N(LP(Ω)) = min {21−1/p, 21/p}, Ω being a σ-finite measure space. The computation of N(Lp) lets us also calculate some other coefficients related to the normal structure.

Vector-valued Littlewood-Paley-Stein Theory for Semigroups II

2018 ◽  
Vol 2020 (21) ◽  
pp. 7769-7791 ◽  
Author(s):  
Quanhua Xu
Keyword(s):  
Banach Space ◽  
Recent Work ◽  
Measure Space ◽  
Convex Banach Space ◽  
Discrete Case ◽  
Uniformly Convex Banach Space ◽  
Uniformly Convex ◽  
Vector Valued ◽  
Markovian Operator ◽  
Symmetric Diffusion

Abstract Inspired by a recent work of Hytönen and Naor, we solve a problem left open in our previous work joint with Martínez and Torrea on the vector-valued Littlewood-Paley-Stein theory for symmetric diffusion semigroups. We prove a similar result in the discrete case, namely, for any $T$ which is the square of a symmetric diffusion Markovian operator on a measure space $(\Omega , \mu )$. Moreover, we show that $T\otimes{ \textrm{Id}}_X$ extends to an analytic contraction on $L_p(\Omega ; X)$ for any $1<p<\infty $ and any uniformly convex Banach space $X$.

A Generalization of a Fixed Point Theorem of Goebel, Kirk and Shimi

1976 ◽  
Vol 19 (1) ◽  
pp. 7-12 ◽  
Author(s):  
Joseph Bogin
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Convex Subset ◽  
Compact Convex ◽  
Continuous Mapping ◽  
Normal Structure ◽  
Convex Banach Space ◽  
Uniformly Convex ◽  
Weakly Compact ◽  
Image Position

In [7], Goebel, Kirk and Shimi proved the following:Theorem. Let X be a uniformly convex Banach space, K a nonempty bounded closed and convex subset of X, and F:K→K a continuous mapping satisfying for each x, y∈K:(1)where ai≥0 and Then F has a fixed point in K.In this paper we shall prove that this theorem remains true in any Banach space X, provided that K is a nonempty, weakly compact convex subset of X and has normal structure (see Definition 1 below).

scholarly journals Fixed points of mappings satisfying semicontractivity conditions

1992 ◽  
Vol 53 (1) ◽  
pp. 25-38
Author(s):  
Jürgen Schu
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Fixed Points ◽  
Convex Subset ◽  
Convex Banach Space ◽  
Uniformly Convex Banach Space ◽  
Uniformly Convex ◽  
Opial’S Condition ◽  
Asymptotically Nonexpansive

AbstractLet A be a subset of a Banach space E. A mapping T: A →A is called asymptoically semicontractive if there exists a mapping S: A×A→A and a sequence (kn) in [1, ∞] such that Tx=S(x, x) for all x ∈A while for each fixed x ∈A, S(., x) is asymptotically nonexpansive with sequence (kn) and S(x,.) is strongly compact. Among other things, it is proved that each asymptotically semicontractive self-mpping T of a closed bounded and convex subset A of a uniformly convex Banach space E which satisfies Opial's condition has a fixed point in A, provided s has a certain asymptoticregurity property.

scholarly journals Classes of operators on vector valued integration spaces

1977 ◽  
Vol 24 (2) ◽  
pp. 129-138 ◽  
Author(s):  
R. J. Fleming ◽  
J. E. Jamison
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Measure Space ◽  
Separable Hilbert Space ◽  
Hermitian Operator ◽  
Finite Measure ◽  
Finite Measure Space ◽  
Image Position ◽  
Vector Valued ◽  
Uniformly Bounded

AbstractLet Lp(Ω, K) denote the Banach space of weakly measurable functions F defined on a finite measure space and taking values in a separable Hilbert space K for which ∥ F ∥p = ( ∫ | F(ω) |p)1/p < + ∞. The bounded Hermitian operators on Lp(Ω, K) (in the sense of Lumer) are shown to be of the form , where B(ω) is a uniformly bounded Hermitian operator valued function on K. This extends the result known for classical Lp spaces. Further, this characterization is utilized to obtain a new proof of Cambern's theorem describing the surjective isometries of Lp(Ω, K). In addition, it is shown that every adjoint abelian operator on Lp(Ω, K) is scalar.

scholarly journals Approximating fixed points by ishikawa iterates

1989 ◽  
Vol 40 (1) ◽  
pp. 113-117 ◽  
Author(s):  
M. Maiti ◽  
M.K. Ghosh
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Fixed Points ◽  
Nonexpansive Mappings ◽  
Convex Banach Space ◽  
Uniformly Convex Banach Space ◽  
Uniformly Convex

In a uniformly convex Banach space the convergence of Ishikawa iterates to a fixed point is discussed for nonexpansive and generalised nonexpansive mappings.

Asymptotic centres of filters on uniformly rotund spaces

1976 ◽  
Vol 15 (1) ◽  
pp. 87-96
Author(s):  
John Staples
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Metric Space ◽  
Fixed Point Theorems ◽  
Convex Banach Space ◽  
Bounded Sequence ◽  
Uniform Convexity ◽  
Uniformly Convex Banach Space ◽  
Uniformly Convex ◽  
Filter Base

The notion of asymptotic centre of a bounded sequence of points in a uniformly convex Banach space was introduced by Edelstein in order to prove, in a quasi-constructive way, fixed point theorems for nonexpansive and similar maps.Similar theorems have also been proved by, for example, adding a compactness hypothesis to the restrictions on the domain of the maps. In such proofs, which are generally less constructive, it may be possible to weaken the uniform convexity hypothesis.In this paper Edelstein's technique is extended by defining a notion of asymptotic centre for an arbitrary set of nonempty bounded subsets of a metric space. It is shown that when the metric space is uniformly rotund and complete, and when the set of bounded subsets is a filter base, this filter base has a unique asymptotic centre. This fact is used to derive, in a uniform way, several fixed point theorems for nonexpansive and similar maps, both single-valued and many-valued.Though related to known results, each of the fixed point theorems proved is either stronger than the corresponding known result, or has a compactness hypothesis replaced by the assumption of uniform convexity.

The Banach-Saks Theorem in C(S)

1974 ◽  
Vol 26 (1) ◽  
pp. 91-97 ◽  
Author(s):  
Nicholas R. Farnum
Keyword(s):  
Banach Space ◽  
Convex Banach Space ◽  
Uniformly Convex Banach Space ◽  
Uniformly Convex

A Banach space X has the Banach-Saks property if every sequence (xn) in X converging weakly to x has a subsequence (xnk) with (1/p)Σk=1xnk converging in norm to x. Originally, Banach and Saks [2] proved that the spaces Lp (p > 1) have this property. Kakutani [4] generalized their result by proving this for every uniformly convex Banach space, and in [9] Szlenk proved that the space L1 also has this property.

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