The kadec-klee property in symmetric spaces of measurable operators

Chilin, Vladimir; Dodds, P. G.; Sukochev, Fedor

doi:10.1007/bf02774037

Cited by 44 publications

(60 citation statements)

References 11 publications

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“…A symmetric space E is said to be K -order continuous (shortly E ∈ (K OC)) if any element x of E is a point of K -order continuity. We refer the reader for more information to see [2,4,5,10].…”

Section: Preliminariesmentioning

confidence: 99%

“…It is necessary to recall the significant paper [2], in which there has been shown a correlation between strict K -monotonicity and global convergence in measure of a sequence of the decreasing rearrangements in symmetric spaces. The next intention of this paper is to find a local version of a correspondence between strict K -monotonicity and global convergence in measure of a sequence of the maximal functions in symmetric spaces.…”

Section: Introductionmentioning

confidence: 99%

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Strict K-monotonicity and K-order continuity in symmetric spaces

Ciesielski

2017

Positivity

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This paper is devoted to strict K -monotonicity and K -order continuity in symmetric spaces. Using a local approach to the geometric structure in a symmetric space E we investigate a connection between strict K -monotonicity and global convergence in measure of a sequence of the maximal functions. Next, we solve an essential problem whether an existence of a point of K -order continuity in a symmetric space E on [0, ∞) implies that the embedding E → L 1 [0, ∞) does not hold. We present a complete characterization of an equivalent condition to K -order continuity in a symmetric space E using a notion of order continuity and the fundamental function of E. We also investigate a relationship between strict K -monotonicity and K -order continuity in symmetric spaces and show some examples of Lorentz spaces and Marcinkiewicz spaces having these properties or not. Finally, we discuss a local version of a crucial correspondence between order continuity and the Kadec-Klee property for global convergence in measure in a symmetric space E.

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Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Strict K-monotonicity and K-order continuity in symmetric spaces

Ciesielski

2017

Positivity

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“…By Corollary 1.3 [10] it follows that λ φ has the Kadec Klee property for global convergence in measure. Since Λ φ satisfies Fatou property [27] we have Λ φ has also semi-Fatou property and by Theorem 3.12 we complete the proof.…”

Section: By the Fatou Property Limmentioning

confidence: 83%

Best approximation properties in spaces of measurable functions

Ciesielski

Lewicki

2017

Mathematische Nachrichten

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Abstract. We research proximinality of µ-sequentially compact sets and µ-compact sets in measurable function spaces. Next we show a correspondence between the Kadec-Klee property for convergence in measure and µ-compactness of the sets in Banach function spaces. Also the property S is investigated in Fréchet spaces and employed to provide the Kadec-Klee property for local convergence in measure. We discuss complete criteria for continuity of metric projection in Fréchet spaces with respect to the Hausdorff distance. Finally, we present the necessary and sufficient condition for continuous metric selection onto a one-dimensional subspace in sequence Lorentz spaces d(w, 1). Mathematics Subjects Classification: 41A65, 46E30, 46A40.

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“…Note that the point (ii) below has been proved directly in [6 (ii) Suppose φ 0 + = 0. Then φ ∈ H g that is each x ∈ φ is an H g -point.…”

Section: Remark 415mentioning

confidence: 94%

“…It is known that the following useful equivalence is true in order continuous symmetric Banach function space: x n − x → 0 if and only if x n → x globally in measure and x * n − x * → 0 (see Corollary 1.6 in [6]). Later the same result has been proved under a weaker assumption that x ∈ E a ([10, Proposition 2.4]).…”

Section: Remark 413mentioning

confidence: 99%