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1.
It is shown that for every 2-concave Banach lattice X of measurable fuctions on the circle, the quotient space X/XA has cotype 2. Here XA denotes the subclass of X consisting of the boundary values of analytic functions. It is also shown that, under slight additional
assumptions, a p-concave operator defined on XA factors through L
A
p
= Hp and extends to X, provided that X is p-convex. Bibliography: 10 titles.
__________
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 333, 2006, pp. 5–16. 相似文献
2.
Given an operator T : X → Y between Banach spaces, and a Banach lattice E consisting of measurable functions, we consider the point-wise extension of the operator to the vector-valued Banach lattices
T
E
: E(X) → E(Y) given by T
E
(f)(ω) = T(f(ω)). It is proved that for any Banach lattice E which does not contain c
0, the operator T is an isomorphism on a subspace isomorphic to c
0 if and only if so is T
E
. An analogous result for invertible operators on subspaces isomorphic to ℓ
1 is also given. 相似文献
3.
T. S. S. R. K. Rao 《Proceedings Mathematical Sciences》1999,109(1):75-85
For 1 ≤p ≤ ∞ we show that there are no denting points in the unit ball of ℓ(lp). This extends a result recently proved by Grząślewicz and Scherwentke whenp = 2 [GS1]. We also show that for any Banach spaceX and for any measure space (Ω, A, μ), the unit ball of ℓ(L
1 (μ), X) has denting points iffL
1(μ) is finite dimensional and the unit ball ofX has a denting point. We also exhibit other classes of Banach spacesX andY for which the unit ball of ℓ(X, Y) has no denting points. When X* has the extreme point intersection property, we show that all ‘nice’ operators in the unit
ball of ℓ(X, Y) are strongly extreme points. 相似文献
4.
Let [A, a] be a normed operator ideal. We say that [A, a] is boundedly weak*-closed if the following property holds: for all Banach spaces X and Y, if T: X → Y** is an operator such that there exists a bounded net (T
i
)
i∈I
in A(X, Y) satisfying lim
i
〈y*, T
i
x
y*〉 for every x ∈ X and y* ∈ Y*, then T belongs to A(X, Y**). Our main result proves that, when [A, a] is a normed operator ideal with that property, A(X, Y) is complemented in its bidual if and only if there exists a continuous projection from Y** onto Y, regardless of the Banach space X. We also have proved that maximal normed operator ideals are boundedly weak*-closed but, in general, both concepts are different.
相似文献
5.
Theω′-topology on the spaceL(X, Y) of bounded linear operators from the Banach spaceX into the Banach spaceY is discussed in [10]. Let ℒw' (X, Y) denote the space of allT∈L(X, Y) for which there exists a sequence of compact linear operators (T
n)⊂K(X, Y) such thatT=ω′−limnTn and let
. We show that
is a Banach ideal of operators and that the continuous dual spaceK(X, Y)* is complemented in
. This results in necessary and sufficient conditions forK(X, Y) to be reflexive, whereby the spacesX andY need not satisfy the approximation property. Similar results follow whenX andY are locally convex spaces.
Financial support from the Potchefstroom University and Maseno University is greatly acknowledged.
Financial support from the NRF and Potchefstroom University is greatly acknowledged. 相似文献
6.
We give sufficient conditions on Banach spaces X and Y so that their projective tensor product X ⊗π
Y, their injective tensor product X ⊗ɛ
Y, or the dual (X ⊗π
Y)* contain complemented copies of ℓp. 相似文献
7.
LetX andY denote two complex Banach spaces and letB(Y, X) denote the algebra of all bounded linear operators fromY toX. ForA∈B(X)
n
,B∈B(Y)
n
, the elementary operator acting onB(Y, X) is defined by
. In this paper we obtain the formulae of the spectrum and the essential spectrum of Δ(A, B) by using spectral mapping theorems. Forn=1, we prove thatS
p
(L
A
,R
B
)=σ(A)×σ(B) and
. 相似文献
8.
Let X be a Banach space with the Grothendieck property, Y a reflexive Banach space, and let X ⊗̌ɛ
Y be the injective tensor product of X and Y.
(a) |
If either X** or Y has the approximation property and each continuous linear operator from X* to Y is compact, then X ⊗̌ɛ
Y has the Grothendieck property. 相似文献
9.
Meng-Kuang Kuo 《印度理论与应用数学杂志》2010,41(6):737-744
The bounds LX,Y (A) and ‖A‖
X,Y
of an operator A = (a
n,k
)
n, k
≥0 with monotonic rows are evaluated, where X and Y are quasi-normed real valued sequence spaces. In particular, in the case where X = ℓ
p
and Y = ℓ
q
, our results give the results when part 0 < p ≤ 1 and 0 < q < ∞ to complement the other results with range p ≥ 1 and 0 < q < ∞. Moreover, we give a partial answer to a problem [1, Problem 7.23] which was posed by Bennett. 相似文献
10.
Let X and Y be Banach spaces. A set
(the space of all weakly compact operators from X into Y) is weakly equicompact if, for every bounded sequence (x
n) in X, there exists a subsequence (x
k(n)) so that (Txk(n)) is uniformly weakly convergent for T ∈ M. In this paper, the notion of weakly equicompact set is used to obtain characterizations of spaces X such that
X ↩̸ ℓ1, of spaces X such that B
X*
is weak* sequentially compact and also to obtain several results concerning to the weak operator and the strong operator
topologies. As another application of weak equicompactness, we conclude a characterization of relatively compact sets in
when this space is endowed with the topology of uniform convergence on the class of all weakly null sequences. Finally, we
show that similar arguments can be applied to the study of uniformly completely continuous sets.
Received: 5 July 2006 相似文献
11.
Raffaella Cilia Joaquín M. Gutiérrez 《Bulletin of the Brazilian Mathematical Society》2009,40(3):371-380
Given real Banach spaces X and Y, let C
wbu1(X, Y) be the space, introduced by R.M. Aron and J.B. Prolla, of C
1 mappings from X into Y such that the mappings and their derivatives are weakly uniformly continuous on bounded sets. We show that f ∈ C
wbu1(X, Y) if and only if f may be written in the form f = g ∘ S, where the intermediate space is normed, S is a precompact operator, and g is a Gateaux differentiable mapping with some additional properties. 相似文献
12.
13.
Let 1 ≤ p < ∞. We show that , the Fremlin projective tensor product of ℓp with a Banach lattice X, has the Radon–Nikodym property if and only if X has the Radon–Nikodym property; and that , the Wittstock injective tensor product of ℓp with a Banach lattice X, has the Radon–Nikodym property if and only if X has the Radon–Nikodym property and each positive operator from ℓp' to X is compact, where 1/p +1/p'= 1 and let ℓp' = c0 if p = 1.
The author gratefully acknowledges support from the Office of Naval Research Grant # N00014-03-1-0621 相似文献
14.
T. S. S. R. K. Rao 《Proceedings Mathematical Sciences》1999,109(3):309-315
In this note we consider the property of being constrained in the bidual, for the space of Bochner integrable functions. For
a Banach spaceX having the Radon-Nikodym property and constrained in its bidual and forY ⊂ X, under a natural assumption onY, we show thatL
1 (μ, X/Y) is constrained in its bidual andL
1 (μ, Y) is a proximinal subspace ofL
1(μ, X). As an application of these results, we show that, ifL
1(μ, X) admits generalized centers for finite sets and ifY ⊂ X is reflexive, thenL
1
μ, X/Y) also admits generalized centers for finite sets. 相似文献
15.
M. D. Voisei 《Set-Valued Analysis》2008,16(4):461-476
This paper is primarily concerned with the problem of maximality for the sum A + B and composition L*
ML in non-reflexive Banach space settings under qualifications constraints involving the domains of A, B, M. Here X, Y are Banach spaces with duals X*, Y*, A, B: X ⇉ X*, M: Y ⇉ Y* are multi-valued maximal monotone operators, and L: X → Y is linear bounded. Based on the Fitzpatrick function, new characterizations for the maximality of an operator as well as
simpler proofs, improvements of previously known results, and several new results on the topic are presented.
相似文献
16.
Let φ be an Orlicz function that has a complementary function φ* and let ℓφ be an Orlicz sequence space. We prove two results in this paper. Result 1:
, the Fremlin projective tensor product of ℓφ with a Banach lattice X, has the Radon-Nikodym property if and only if both ℓφ and X have the Radon-Nikodym property. Result 2:
, the Wittstock injective tensor product of ℓφ with a Banach lattice X, has the Radon-Nikodym property if and only if both ℓφ and X have the Radon-Nikodym property and each positive continuous linear operator from hφ* to X is compact.
We dedicate this paper to the memory of H. H. Schaefer
The first-named author gratefully acknowledges support from the Faculty Research Program of the University of Mississippi
in summer 2004. 相似文献
17.
K. E. Tikhomirov 《Siberian Mathematical Journal》2011,52(1):147-158
We give a criterion for the uniform relative K-monotonicity of weighted couples (X,X(w
1)) and (X,X(w
2)), where X is some Banach lattice of measurable functions with the Fatou property while w
1 and w
2 are weight functions. Using the criterion, we prove some corollaries for sequence spaces and arbitrary Banach lattices. 相似文献
18.
C. Alegre 《Acta Mathematica Hungarica》2009,122(4):357-372
If (X, p) and (Y, q) are two asymmetric normed spaces, the set LC(X, Y) of all continuous linear mappings from (X, p) to (Y, q) is not necessarily a linear space, it is a cone. If X and Y are two Banach lattices and p and q are, respectively, their associated asymmetric norms (p(x) = ‖+‖, q(y) = ‖y
+‖), we prove that the positive operators from X to Y are elements of the cone LC(X, Y). We also study the dual space of an asymmetric normed space and finally we give open mapping and closed graph type theorems
in the framework of asymmetric normed spaces. The classical results for normed spaces follow as particular cases.
The author acknowledges the support of the Ministerio de Educación y Ciencia of Spain and FEDER, under grant MTM2006-14925-C02-01
and Generalitat Valenciana under grant GV/2007/198. 相似文献
19.
《Quaestiones Mathematicae》2013,36(3-4):269-288
Abstract Using a lifting of £∞ (μ, X) ([5],[6]), we construct a lifting ρ x of the seminormed vector space £∞ (μ, X) of measurable, essentially bounded X-valued functions. We show that in a certain sense such a lifting always exists. If μ is Lebesgue measure on (0, 1) we show that ρ x exists as map from £∞ ((O, 1), X) → £∞,((0, l), X) if and only if X is reflexive. In general the lifted function takes its values in X **. Therefore we investigate the question, when f ε £∞ (μ, X) is strictly liftable in the sense that the lifted function is a map with values even in X. As an application we introduce the space £∞ strong (μ, L (X, Y**)), a subspace of the space of strongly measurable, essentially bounded L (X, Y, **)-valued functions, and the associated quotient space £∞ strong (μ, L (X,Y**)). We show that this space is a Banach space because there is a kind of a Dunford-Pettis Theorem for a subspace of L (X, £∞(μ Y**)). Finally we investigate the measurability property of functions in £∞(μ Y**)) und see that there exists a connection to the Radon-Nikodym property of the space L (X, Y). 相似文献
20.
An elementary proof of the (known) fact that each element of the Banach spaceℓ
w
p
(X) of weakly absolutelyp-summable sequences (if 1≤p<∞) in the Banach spaceX is the norm limit of its sections if and only if each element ofℓ
w
p
(X) is a norm null sequence inX, is given. Little modification to this proof leads to a similar result for a family of Orlicz sequence spaces. Some applications
to spaces of compact operators on Banach sequence spaces are considered. 相似文献
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