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1
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Complex rotundities and midpoint local uniform rotundity in symmetric spaces of measurable operators

100%
Czerwińska M., Kamińska A.
Studia Mathematica
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2010
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tom 201
|
nr 3
253-285
EN
We investigate the relationships between strongly extreme, complex extreme, and complex locally uniformly rotund points of the unit ball of a symmetric function space or a symmetric sequence space E, and of the unit ball of the space E(ℳ,τ) of τ-measurable operators associated to a semifinite von Neumann algebra (ℳ,τ) or of the unit ball in the unitary matrix space $C_{E}$. We prove that strongly extreme, complex extreme, and complex locally uniformly rotund points x of the unit ball of the symmetric space E(ℳ,τ) inherit these properties from their singular value function μ(x) in the unit ball of E with additional necessary requirements on x in the case of complex extreme points. We also obtain the full converse statements for the von Neumann algebra ℳ with a faithful, normal, σ-finite trace τ as well as for the unitary matrix space $C_{E}$. Consequently, corresponding results on the global properties such as midpoint local uniform rotundity, complex rotundity and complex local uniform rotundity follow.
2

Banach envelopes of some quasi-Banach function spaces

100%
Kamińska A., Kubiak D.
Commentationes Mathematicae
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2016
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tom 56
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nr 1
EN
We show that under some assumptions on the Musielak−Orlicz function generating a quasi-Banach Musielak−Orlicz function space, the Banach envelope of the weighted Cesàro−Musielak−Orlicz space generated by a certain positive sublinear operator is a weighted \(L_1\)-space.
3

Asymptotically isometric and isometric copies of \(\ell_1\) in some Banach function lattices

100%
Kamińska A., ~Mastyło M.
Commentationes Mathematicae
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2013
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tom 53
|
nr 2
EN
We identify the class of Caldern-Lozanovskii spaces that do not contain an asymptotically isometric copy of \(\ell_1\), and consequently we obtain the corresponding characterizations in the classes of Orlicz-Lorentz and Orlicz spaces equipped with the Luxemburg norm. We also give a~complete description of order continuous Orlicz-Lorentz spaces which contain (order) isometric copies of \(\ell_1^{(n)}\) for each integer \(n \geq 2\).~As an application we provide necessary and sufficient conditions for order continuous Orlicz-Lorentz spaces to contain an (order) isometric copy of \(\ell_1\).~In particular we give criteria in Orlicz and Lorentz spaces for (order) isometric containment of \(\ell_1^{(n)}\) and \(\ell_1\).~The results are applied to obtain the~description of universal Orlicz-Lorentz spaces for all two-dimensional normed spaces.
4
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Order convexity and concavity of Lorentz spaces $Λ_{p,w}$, 0 < p < ∞

100%
Kamińska A., Maligranda L.
Studia Mathematica
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2004
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tom 160
|
nr 3
267-286
EN
We study order convexity and concavity of quasi-Banach Lorentz spaces $Λ_{p,w}$, where 0 < p < ∞ and w is a locally integrable positive weight function. We show first that $Λ_{p,w}$ contains an order isomorphic copy of $l^{p}$. We then present complete criteria for lattice convexity and concavity as well as for upper and lower estimates for $Λ_{p,w}$. We conclude with a characterization of the type and cotype of $Λ_{p,w}$ in the case when $Λ_{p,w}$ is a normable space.
5
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Dual spaces to Orlicz-Lorentz spaces

81%
Kamińska A., Leśnik K., Raynaud Y.
Studia Mathematica
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2014
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tom 222
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nr 3
229-261
EN
For an Orlicz function φ and a decreasing weight w, two intrinsic exact descriptions are presented for the norm in the Köthe dual of the Orlicz-Lorentz function space $Λ_{φ,w}$ or the sequence space $λ_{φ,w}$, equipped with either the Luxemburg or Amemiya norms. The first description is via the modular $inf{∫ φ⁎(f*/|g|)|g|: g ≺ w}$, where f* is the decreasing rearrangement of f, ≺ denotes submajorization, and φ⁎ is the complementary function to φ. The second description is in terms of the modular $∫_{I} φ⁎((f*)⁰/w)w$,where (f*)⁰ is Halperin's level function of f* with respect to w. That these two descriptions are equivalent results from the identity $inf{ ∫ ψ(f*/|g|)|g|: g ≺ w} = ∫_{I} ψ((f*)⁰/w)w$, valid for any measurable function f and any Orlicz function ψ. An analogous identity and dual representations are also presented for sequence spaces.
6
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Complex Convexity of Orlicz-Lorentz Spaces and its Applications

81%
Choi C., Kamińska A., Lee H.
Bulletin of the Polish Academy of Sciences. Mathematics
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2004
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tom 52
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nr 1
19-38
EN
We give sufficient and necessary conditions for complex extreme points of the unit ball of Orlicz-Lorentz spaces, as well as we find criteria for the complex rotundity and uniform complex rotundity of these spaces. As an application we show that the set of norm-attaining operators is dense in the space of bounded linear operators from $d*_(w,1)$ into d(w,1), where $d*_(w,1)$ is a predual of a complex Lorentz sequence space d(w,1), if and only if wi ∈ c₀∖ℓ₂.
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