Uniform Kadec–Klee–Huff properties of vector-valued Hardy spaces

1993 ◽  
Vol 114 (1) ◽  
pp. 25-30 ◽  
Author(s):  
P. N. Dowling ◽  
C. J. Lennard
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Hardy Spaces ◽  
Weak Topology ◽  
Uniformly Convex ◽  
Vector Valued

AbstractIn [8] Partington showed that a Banach space X is uniformly convex if and only if Lp([0, 1], X) has the uniform Kadec–Klee–Huff property with respect to the weak topology (UKKH (weak)), where 1 < p < ∞. In this note we will characterize the Banach spaces X such that HP(D, X) has UKKH (weak), where 1 ≤ p < ∞. Similar results for UKKH (weak*) are also obtained. These results (and proofs) are quite different from Partington's result (and proof).

Kadec–Klee properties of vector-valued Hardy spaces

1992 ◽  
Vol 111 (3) ◽  
pp. 535-544 ◽  
Author(s):  
P. N. Dowling ◽  
C. J. Lennard
Keyword(s):  
Banach Space ◽  
Uniform Convergence ◽  
Hardy Spaces ◽  
Unit Disc ◽  
Open Unit ◽  
Uniformly Convex ◽  
Open Unit Disc ◽  
Vector Valued ◽  
Sequence Of Functions ◽  
Compact Subsets

In 1930, S. Warschawski [19] showed that H1(D), where D is the open unit disc in ℂ, has the following property: Let be a sequence of functions in H1(D) converging uniformly on compact subsets of D to a function f∈H1(D) and suppose that ‖fn‖1 = |f‖1 = 1 for all n∈ℕ. Then converges to zero. From a Banach space standpoint, this result says that H1(D) has the Kadec–Klee property with respect to uniform convergence on compact subsets of D. Warschawski's result was proved independently by Newman [16] in 1963 (see also [13] for another proof) and extended to more general domains by Hoffman [12], Goldstein and Swaminathan [8] and Godefroy [7]. A uniform version of Warschawski's result and its subsequent extensions was recently obtained by Besbes, Dilworth, Dowling and Lennard [2] (see also [1]). We mention here that these results for H1 spaces also hold for the Hp-spaces for 1 < p < ∞ because these spaces are uniformly convex.

scholarly journals A note on best approximation and invertibility of operators on uniformly convex Banach spaces

1991 ◽  
Vol 14 (3) ◽  
pp. 611-614 ◽  
Author(s):  
James R. Holub
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Banach Spaces ◽  
Best Approximation ◽  
Bounded Linear Operator ◽  
Convex Banach Space ◽  
Sufficient Condition ◽  
Uniformly Convex ◽  
Bounded Linear ◽  
Uniformly Convex Banach Spaces

It is shown that ifXis a uniformly convex Banach space andSa bounded linear operator onXfor which‖I−S‖=1, thenSis invertible if and only if‖I−12S‖<1. From this it follows that ifSis invertible onXthen either (i)dist(I,[S])<1, or (ii)0is the unique best approximation toIfrom[S], a natural (partial) converse to the well-known sufficient condition for invertibility thatdist(I,[S])<1.

On Additive Operators

1971 ◽  
Vol 23 (3) ◽  
pp. 468-480 ◽  
Author(s):  
N. A. Friedman ◽  
A. E. Tong
Keyword(s):  
Banach Space ◽  
Compact Hausdorff Space ◽  
Weak Topology ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
Additive Functional ◽  
Representation Theorems ◽  
Image Position ◽  
Vector Valued ◽  
Kernel Representation

Representation theorems for additive functional have been obtained in [2, 4; 6-8; 10-13]. Our aim in this paper is to study the representation of additive operators.Let S be a compact Hausdorff space and let C(S) be the space of real-valued continuous functions defined on S. Let X be an arbitrary Banach space and let T be an additive operator (see § 2) mapping C(S) into X. We will show (see Lemma 3.4) that additive operators may be represented in terms of a family of “measures” {μh} which take their values in X**. If X is weakly sequentially complete, then {μh} can be shown to take their values in X and are vector-valued measures (i.e., countably additive in the norm) (see Lemma 3.7). And, if X* is separable in the weak-* topology, T may be represented in terms of a kernel representation satisfying the Carathéordory conditions (see [9; 11; §4]):

scholarly journals Multipliers on Vector Valued Bergman Spaces

2002 ◽  
Vol 54 (6) ◽  
pp. 1165-1186 ◽  
Author(s):  
Oscar Blasco ◽  
José Luis Arregui
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Bergman Space ◽  
Unit Disc ◽  
Bergman Spaces ◽  
Complex Banach Space ◽  
Bounded Operators ◽  
Taylor Coefficients ◽  
Vector Valued

AbstractLet X be a complex Banach space and let Bp(X) denote the vector-valued Bergman space on the unit disc for 1 ≤ p < ∞. A sequence (Tn)n of bounded operators between two Banach spaces X and Y defines a multiplier between Bp(X) and Bq(Y) (resp. Bp(X) and lq(Y)) if for any function we have that belongs to Bq(Y) (resp. (Tn(xn))n ∈ lq(Y)). Several results on these multipliers are obtained, some of them depending upon the Fourier or Rademacher type of the spaces X and Y. New properties defined by the vector-valued version of certain inequalities for Taylor coefficients of functions in Bp(X) are introduced.

scholarly journals On Best Proximity Results for a Generalized Modified Ishikawa’s Iterative Scheme Driven by Perturbed 2-Cyclic Like-Contractive Self-Maps in Uniformly Convex Banach Spaces

2019 ◽  
Vol 2019 ◽  
pp. 1-15 ◽  
Author(s):  
M. De la Sen ◽  
Mujahid Abbas
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Iterative Process ◽  
Iterative Scheme ◽  
Convex Banach Space ◽  
The Self ◽  
Uniformly Convex ◽  
Cyclic Structure ◽  
Best Proximity Points ◽  
Uniformly Convex Banach Spaces

This paper proposes a generalized modified iterative scheme where the composed self-mapping driving can have distinct step-dependent composition order in both the auxiliary iterative equation and the main one integrated in Ishikawa’s scheme. The self-mapping which drives the iterative scheme is a perturbed 2-cyclic one on the union of two sequences of nonempty closed subsets Ann=0∞ and Bnn=0∞ of a uniformly convex Banach space. As a consequence of the perturbation, such a driving self-mapping can lose its cyclic contractive nature along the transients of the iterative process. These sequences can be, in general, distinct of the initial subsets due to either computational or unmodeled perturbations associated with the self-mapping calculations through the iterative process. It is assumed that the set-theoretic limits below of the sequences of sets Ann=0∞ and Bnn=0∞ exist. The existence of fixed best proximity points in the set-theoretic limits of the sequences to which the iterated sequences converge is investigated in the case that the cyclic disposal exists under the asymptotic removal of the perturbations or under its convergence of the driving self-mapping to a limit contractive cyclic structure.

scholarly journals Generalized Projections on Closed Nonconvex Sets in Uniformly Convex and Uniformly Smooth Banach Spaces

2015 ◽  
Vol 2015 ◽  
pp. 1-7 ◽  
Author(s):  
Messaoud Bounkhel
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Convex Set ◽  
Variational Problems ◽  
Generalized Projection ◽  
Uniformly Convex ◽  
Uniformly Smooth ◽  
Nonconvex Sets ◽  
Generalized Projections ◽  
Uniformly Smooth Banach Spaces

The present paper is devoted to the study of the generalized projectionπK:X∗→K, whereXis a uniformly convex and uniformly smooth Banach space andKis a nonempty closed (not necessarily convex) set inX. Our main result is the density of the pointsx∗∈X∗having unique generalized projection over nonempty close sets inX. Some minimisation principles are also established. An application to variational problems with nonconvex sets is presented.

On optimal mild solutions of non-homogeneous differential equations in Banach spaces

1979 ◽  
Vol 84 (3-4) ◽  
pp. 273-277 ◽  
Author(s):  
S. Zaidman
Keyword(s):  
Banach Space ◽  
Differential Equations ◽  
Banach Spaces ◽  
Real Line ◽  
Infinitesimal Generator ◽  
Existence Result ◽  
Mild Solutions ◽  
Optimal Solutions ◽  
Uniformly Convex ◽  
Optimal Mild Solutions

SynopsisConsider mild solutions on the real line of non-homogeneous differential equations in a Banach space: u′(t) = Au(t) + f(t), where A is the infinitesimal generator of a C0-semigroup.We prove an existence result for optimal solutions (as defined in the text) in reflexive spaces and an uniqueness fact in uniformly convex B-spaces.

scholarly journals Algorithm for Hammerstein equations with monotone mappings in certain Banach spaces

2016 ◽  
Vol 25 (1) ◽  
pp. 107-120
Author(s):  
T. M. M. SOW ◽  
◽  
C. DIOP ◽  
N. DJITTE ◽  
◽  
...  
Keyword(s):  
Banach Space ◽  
Strong Convergence ◽  
Banach Spaces ◽  
Iterative Process ◽  
Real Banach Space ◽  
Monotone Mappings ◽  
Uniformly Convex ◽  
Hammerstein Equation ◽  
Uniformly Smooth ◽  
Monotone Maps

For q > 1 and p > 1, let E be a 2-uniformly convex and q-uniformly smooth or p- uniformly convex and 2-uniformly smooth real Banach space and F : E → E∗, K : E∗ → E be bounded and strongly monotone maps with D(K) = R(F) = E∗. We construct a coupled iterative process and prove its strong convergence to a solution of the Hammerstein equation u + KF u = 0. Futhermore, our technique of proof is of independent of interest.

SIP Xd-frames and their perturbations in uniformly convex Banach spaces

2014 ◽  
Vol 12 (03) ◽  
pp. 1450024
Author(s):  
Xianwei Zheng ◽  
Shouzhi Yang
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Separable Banach Space ◽  
Inner Product ◽  
Uniformly Convex ◽  
Atomic Decompositions ◽  
Banach Frames ◽  
Uniformly Convex Banach Spaces ◽  
Bk Space ◽  
The Relationship

In this paper, we introduce the definitions of SIP-I and SIP-II Xd-frames in a uniformly convex, separable Banach space X with respect to a BK-space Xd (here SIP represents semi-inner product), both of them are defined as sequence of elements in X. We characterize SIP-I and SIP-II Xd-frames in terms of the corresponding synthesis and analysis operators, respectively, then we consider perturbations for both of them. Furthermore, we also introduce the definitions of SIP Banach frames and SIP atomic decompositions. Under certain assumptions, we establish the relationship between SIP Banach frames and SIP atomic decompositions, and therefore obtain reconstruction formulas for every element in X and X* by using a pair of SIP-I and SIP-II Xd-frames for X. Finally, we discuss perturbations of SIP Banach frames and SIP atomic decompositions.

scholarly journals The fixed-point property in Banach spaces containing a copy ofc0

2003 ◽  
Vol 2003 (3) ◽  
pp. 183-192
Author(s):  
Maria A. Japón Pineda
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Banach Spaces ◽  
Weak Topology ◽  
Nonexpansive Mappings ◽  
Fixed Point Property ◽  
Point Property ◽  
Asymptotically Nonexpansive ◽  
Locally Convex Topology ◽  
Locally Convex

We prove that every Banach space containing an isomorphic copy ofc0fails to have the fixed-point property for asymptotically nonexpansive mappings with respect to some locally convex topology which is coarser than the weak topology. If the copy ofc0is asymptotically isometric, this result can be improved, because we can prove the failure of the fixed-point property for nonexpansive mappings.

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