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1

Copies of the sequence space $\omega$ in $F$-lattices with applications to Musielak−Orlicz spaces

100%

Wójtowicz M., Wiśniewska H.

Commentationes Mathematicae

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2016

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tom 56

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nr 1

EN

Let $E$ be a fixed real function $F$-space, i.e., $E$ is an order ideal in $L_0(S,\Sigma,\mu)$ endowed with a monotone $F$-norm $\|\|$ under which $E$ is topologically complete. We prove that $E$ contains an isomorphic (topological) copy of $\omega$, the space of all sequences, if and only if $E$ contains a lattice-topological copy $W$ of $\omega$. If $E$ is additionally discrete, we obtain a much stronger result: $W$ can be a projection band; in particular, $E$ contains a~complemented copy of $\omega$. This solves partially the open problem set recently by W. Wnuk. The property of containing a copy of $\omega$ by a Musielak−Orlicz space is characterized as follows. (1) A sequence space $\ell_{\Phi}$, where $\Phi = (\varphi_n)$, contains a copy of $\omega$ iff $\inf_{n \in \mathbb{N}} \varphi_n (\infty) = 0$, where $\varphi_n (\infty) = \lim_{t \to \infty} \varphi_n (t)$. (2) If the measure $\mu$ is atomless, then $\omega$ embeds isomorphically into $L_{\mathcal{M}} (\mu)$ iff the function $\mathcal{M}_{\infty}$ is positive and bounded on some set $A\in \Sigma$ of positive and finite measure, where $\mathcal{M}_{\infty} (s) = \lim_{n \to \infty} \mathcal{M} (n, s)$, $s\in S$. In particular, (1)' $\ell_\varphi$ does not contain any copy of $\omega$, and (2)' $L_{\varphi} (\mu)$, with $\mu$ atomless, contains a~copy $W$ of $\omega$ iff $\varphi$ is bounded, and every such copy $W$ is uncomplemented in $L_{\varphi} (\mu)$.

2

The controlled separable projection property for Banach spaces

100%

Ferrer J., Wójtowicz M.

Open Mathematics

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2011

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tom 9

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nr 6

1252-1266

EN

Let X, Y be two Banach spaces. We say that Y is a quasi-quotient of X if there is a continuous operator R: X → Y such that its range, R(X), is dense in Y. Let X be a nonseparable Banach space, and let U, W be closed subspaces of X and Y, respectively. We prove that if X has the Controlled Separable Projection Property (CSPP) (this is a weaker notion than the WCG property) and Y is a quasi-quotient of X, then the structure of Y resembles the structure of a separable Banach space: (a) Y/W is norm-separable iff its dual W ⊥ is weak*-separable, (b) every weak*-separable subset of B Y* is weak*-metrizable, (c) every weak*-null sequence in the unit sphere of Y* contains a “nice“ subsequence; and (d) if U is separable, then X/U also has the CSPP. Property (a) yields an easy way of obtaining separable quotients in a class of Banach spaces. We also study the CSPP for C(K)-spaces, where K is a Mrówka compact space, e.g., we prove that the CSPP is not a three-space property.

3

Cantor-Bernstein theorems for Orlicz sequence spaces

81%

Finol C., González M., Wójtowicz M.

Banach Center Publications

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2014

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tom 102

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nr 1

71-88

EN

For two Banach spaces X and Y, we write $dim_{ℓ}(X) = dim_{ℓ}(Y)$ if X embeds into Y and vice versa; then we say that {X and Y have the same linear dimension}. In this paper, we consider classes of Banach spaces with symmetric bases. We say that such a class ℱ has the Cantor-Bernstein property if for every X,Y ∈ ℱ the condition $dim_{ℓ}(X) = dim_{ℓ}(Y)$ implies the respective bases (of X and Y) are equivalent, and hence the spaces X and Y are isomorphic. We prove (Theorems 3.1, 3.3, 3.5) that the class of Orlicz sequence spaces generated by regularly varying Orlicz functions is of this type. This complements some results in this direction obtained earlier by S. Banach (Proposition 1.1), L. Drewnowski (Proposition 1.2), and M. J. Gonzalez, B. Sari and M. Wójtowicz (Theorem 1.4). Our theorems apply to large families of concrete Orlicz spaces.

Ograniczanie wyników

1 Banach Center Publications

1 Commentationes Mathematicae

1 Open Mathematics

3 Wójtowicz M.

1 Ferrer J.

1 Finol C.

1 González M.

1 Wiśniewska H.

1 2016

1 2014

1 2011

Copies of the sequence space \(\omega\) in \(F\)-lattices with applications to Musielak−Orlicz spaces

The controlled separable projection property for Banach spaces

Cantor-Bernstein theorems for Orlicz sequence spaces