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Measures of noncompactness and normal structure in Banach spaces

100%

García-Falset J., Jiménez-Melado A., Lloréns-Fuster E.

Studia Mathematica

1994

tom 110

nr 1

1-8

Sufficient conditions for normal structure of a Banach space are given. One of them implies reflexivity for Banach spaces with an unconditional basis, and also for Banach lattices.

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On \(\ell_1\)-preduals distant by 1

75%

Piasecki Ł.

Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica

2018

tom 72

nr 2

For every predual \(X\) of \(\ell_1\) such that the standard basis in \(\ell_1\) is weak\(^*\) convergent, we give explicit models of all Banach spaces \(Y\) for which the Banach-Mazur distance \(d(X,Y)=1\). As a by-product of our considerations, we obtain some new results in metric fixed point theory. First, we show that the space \(\ell_1\), with a predual \(X\) as above, has the stable weak\(^*\) fixed point property if and only if it has almost stable weak\(^*\) fixed point property, i.e. the dual \(Y^*\) of every Banach space \(Y\) has the weak\(^*\) fixed point property (briefly, \(\sigma(Y^*,Y)\)-FPP) whenever \(d(X,Y)=1\). Then, we construct a predual \(X\) of \(\ell_1\) for which \(\ell_1\) lacks the stable \(\sigma(\ell_1,X)\)-FPP but it has almost stable \(\sigma(\ell_1,X)\)-FPP, which in turn is a strictly stronger property than the \(\sigma(\ell_1,X)\)-FPP. Finally, in the general setting of preduals of \(\ell_1\), we give a sufficient condition for almost stable weak\(^*\) fixed point property in \(\ell_1\) and we prove that for a wide class of spaces this condition is also necessary.

Ograniczanie wyników

1 Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica

1 Studia Mathematica

1 García-Falset J.

1 Jiménez-Melado A.

1 Lloréns-Fuster E.

1 Piasecki Ł.

1 2018

1 1994

Wyniki wyszukiwania

Measures of noncompactness and normal structure in Banach spaces

On \(\ell_1\)-preduals distant by 1