On property \((\beta)\) in Musielak-Orlicz sequence spaces of Bochner type (Q2752303)

Property (\(\beta\)) is one of several intermediate stages between reflexivity of Banach spaces and uniform convexity of Banach spaces. Recall that a space is uniformly convex if \(\forall \varepsilon>0, \exists \delta>0\), such that if \(x, y \in S(X)\) (the unit sphere in \(X\)), are such that \(\| x - y \| > \varepsilon\), then \(\| { 1 \over 2} (x + y)\| < 1 - \delta\). One form of property (\(\beta\)) is \(\;\forall \varepsilon>0, \;\exists \delta>0\), such that for each element \(x \in B(X)\) (the unit ball in \(X\)), and each sequence \(x_n \in B(X)\) such that \(\inf\{ \| x_n - x_m\| : n \neq m \} \geq \varepsilon\), then there is a \(k\) such that \(\| {1 \over 2} (x + x_k)\| \leq 1 - \delta\). Various other properties are discussed as well (see the paper) such as near uniform convexity (NUC) and the drop property (D). For Banach spaces, one has NEWLINE\[NEWLINE UC \Rightarrow \beta \Rightarrow NUC \Rightarrow D \Rightarrow \;\text{reflexivity} NEWLINE\]NEWLINE where in general the converse implications are false. However, in Orlicz-Lorentz spaces UC and \(\beta\) are equivalent, while in Orlicz sequence spaces, \(\beta\) and reflexivity are equivalent.NEWLINENEWLINELet \(M\) be the space of real sequences. A function \(\phi\) from \(\mathbb{R}\) to the positive real numbers is an Orlicz function if it is convex, even, \(\phi(0) = 0\), but \(\phi\) is not identically equal to zero, and \(\lim_{u \to 0} \phi(u)/u = 0\). A sequence \(\phi = (\phi_i)\) of Orlicz functions is called a Musielak-Orlicz function. A convex modular on \(M\) is NEWLINE\[NEWLINE I_{\phi}(x) = \sum_{i = 1}^{\infty} \phi_i(u_i). NEWLINE\]NEWLINE The Musielak-Orlicz space \(l_{\phi}\) is NEWLINE\[NEWLINE l_{\phi} = \{ x \in M | I_{\phi}(cx) < \infty \;\;\text{for some } c> 0 \}. NEWLINE\]NEWLINE The space can be given the Luxemburg norm NEWLINE\[NEWLINE \| x\| _{\phi} = \inf \{ \varepsilon > 0 | I_{\phi}(\tfrac x \varepsilon) \leq 1 \}NEWLINE\]NEWLINE or the Orlicz (Amemiya) norm NEWLINE\[NEWLINE \| x\| _0 = \inf_{k > 0} \{ \tfrac 1k \left(1 + I_{\phi}(kx) \right) \}. NEWLINE\]NEWLINE Finally, if \(X\) is a Banach space, \(l_{\phi}(X)\) is the space of all sequences \((x_i)\) of elements of \(X\) such that the sequence \((\| x_i\| _X)\) belongs to \(l_{\phi}\).NEWLINENEWLINEThe author considers property (\(\beta\)) in Musielak-Orlicz sequence spaces and shows that \(l_{\phi}(X)\) has property (\(\beta\)) if and only if both spaces \(l_{\phi}\) and \(X\) have it also. He gives results in the case of both the Luxemburg and Orlicz norms.