The uniform Kadec-Klee property for the Lorentz spaces Lw,1

Dilworth, S. J.; Hsu, Yu-Ping

doi:10.1017/s1446788700037356

Cited by 6 publications

(5 citation statements)

References 15 publications

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“…We now recover the result proven in [3] by Dilworth and Hsu. It is a corollary of Theorems 3·9, 3·10 and 3·14, since in that case condition (M) means exactly that w ∈ C 2 .…”

supporting

confidence: 87%

“…It is a corollary of Theorems 3•9, 3•10 and 3•14, since in that case condition (M) means exactly that w ∈ C 2 . (We remark that our slightly stronger conclusion is implicit in the proofs of [3].) 374 A. KAMIŃSKA, C.J.…”

Section: By Passing To a Subsequence We May Assume That For Everymentioning

confidence: 74%

“…Hence, by applying lemma 2·5 in [3] to the functions ϕ(| f |), ϕ(| f 1 |) and ϕ(| f 2 |), there exists α 2 (ε) > 0 such that…”

Section: Proposition 3·1 If E Is a Symmetric Space With The Fatou Prmentioning

confidence: 99%

See 2 more Smart Citations

The uniform Kadec–Klee property for Orlicz–Lorentz spaces

Kamińska¹,

Lennard²,

Mastyło

et al. 2007

Math. Proc. Camb. Phil. Soc.

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“…We now recover the result proven in [3] by Dilworth and Hsu. It is a corollary of Theorems 3·9, 3·10 and 3·14, since in that case condition (M) means exactly that w ∈ C 2 .…”

supporting

confidence: 87%

Section: By Passing To a Subsequence We May Assume That For Everymentioning

confidence: 74%

See 1 more Smart Citation

The uniform Kadec–Klee property for Orlicz–Lorentz spaces

Kamińska¹,

Lennard²,

Mastyło

et al. 2007

Math. Proc. Camb. Phil. Soc.

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“…It is proved there that a necessary and sufficient condition for Λ w to have the Kadec-Klee property is the condition that the weight function w is strictly decreasing. Note also that recently Dilworth and Hsu [5]…”

mentioning

confidence: 87%

On Geometric Properties of Orlicz-Lorentz Spaces

1997

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“…While uniformly convex spaces are necessarily reflexive, it turns out that many classical non-reflexive spaces, e.g. the Hardy space H 1 of analytic functions on the ball or on the polydisc in C N [5], the Lorentz space L p,1 (µ) [12,14] and the trace class operators L 1 (H) [24], satisfy the UKK or the UKK * property.…”

Section: Background and Basic Definitionsmentioning

confidence: 99%

Geometric implications of the M (r, s)-properties and the uniform Kadec-Klee property in JB^*-triples

Nieto

Peralta³

2016

Q J Math

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We explore new implications of the M (r, s) and M * (r, s) properties for Banach spaces. We show that a Banach space X satisfying property M (1, s) for some 0 < s ≤ 1, admitting a point x 0 in its unit sphere at which the relative weak and norm topologies agree, satisfies the generalized Gossez-Lami Dozo property. We establish sufficient conditions, in terms of the (r, s)-Lipschitz weak * Kadec-Klee property on a Banach space X, to guarantee that its dual space satisfies the UKK * property. We determine appropriate conditions to assure that a Banach space X satisfies the (r, s)-Lipschitz weak * Kadec-Klee property. These results are applied to prove that every spin factor satisfies the UKK property, and consequently, the KKP and the UKK properties are equivalent for real and complex JB * -triples.It should be noted here that properties M(1, 1) and M * (1, 1) are precisely properties (M) and (M * ) in the terminology of [23].Properties M(r, s) have been successfully applied in fixed point theory (compare [20, 15, 10]).The fixed point theory motivated the developing of many interesting properties in Banach space theory. That is also the case of the Kadec-Klee (or Radon-Riesz) property (KKP in the sequel). We recall that a Banach space has the KKP if any sequence in the unit sphere whose weak limit is also in the unit sphere, is indeed norm convergent. The uniform Kadec-Klee property for the weak topology on a Banach space was introduced by R. Huff [24] as a useful substitute for uniform convexity, especially in many non-reflexive spaces. D. van Dulst and B. Sims [17]showed that the uniform Kadec-Klee property for weak and weak * topologies implies weak (resp. weak * ) normal structure.It is well-known (see [2,34]) that the Schatten p-classes C p , 1 ≤ p < ∞, have the KKP. It has been shown by C. Lennard [30] that the direct argument given by J. Arazy [2] for trace-class operators can be refined to show that L 1 (H), the space of trace class operators on an arbitrary infinite-dimensional Hilbert space H satisfies a stronger property called the uniformly Kadec-Klee in the weak * topology UKK * (see section 2 for the detailed definitions). From a somewhat different viewpoint, it is a classical theorem of F. Riesz that norm convergence for sequences in the unit sphere of L 1 [0, 1] coincides with convergence in measure. Appropriate uniform version of this theorem may be found in [31].Independently from the fixed point theory, The Kadec-Klee property has been deeply studied in certain particular classes of Banach spaces including C * -algebras and JB * -triples in connection with the Alternative Dunford-Pettis property (compare [1, 6, 3] and [8]). Proposition 2.13 in [6] provides a complete description of those JB * -triples satisfying the KKP, namely, a JB * -triple satisfies this property if and only if it is finite-dimensional or a Hilbert space or a spin factor. A similar conclusion holds for real JB * -triples (compare [3, Proposition 3.13]). It is a natural open problem to ask wether a JB * -triple satisfying t...

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The uniform Kadec-Klee property for the Lorentz spaces L_w,₁

Abstract: In this paper we show that the Lorentz space

Cited by 6 publications

References 15 publications

The uniform Kadec–Klee property for Orlicz–Lorentz spaces

The uniform Kadec–Klee property for Orlicz–Lorentz spaces

On Geometric Properties of Orlicz-Lorentz Spaces

Geometric implications of the M (r, s)-properties and the uniform Kadec-Klee property in JB^*-triples

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The uniform Kadec-Klee property for the Lorentz spaces Lw,1

Abstract: In this paper we show that the Lorentz space

Cited by 6 publications

References 15 publications

The uniform Kadec–Klee property for Orlicz–Lorentz spaces

The uniform Kadec–Klee property for Orlicz–Lorentz spaces

On Geometric Properties of Orlicz-Lorentz Spaces

Geometric implications of the M (r, s)-properties and the uniform Kadec-Klee property in JB*-triples

Contact Info

Product

Resources

About

The uniform Kadec-Klee property for the Lorentz spaces L_w,₁

Geometric implications of the M (r, s)-properties and the uniform Kadec-Klee property in JB^*-triples