Opial's modulus in Köthe sequence spaces (Q2733917)

Let \((X,\|\cdot\|)\) be a real Banach space without Schur's property, i.e. the weak and strong convergence does not coincide for sequences. Let \(X^*\) be a dual space of \(X\). Denote by \(B(X)\) and \(S(X)\) the closed unit ball and the unit sphere of \(X\), respectively. A Banach space \(X\) is said to have the Opial property if every weakly null sequence \((x_n)\subset X\) satisfies \(\liminf_{n\to\infty}\|x_n\|<\liminf_{n\to\infty}\|x_n+x\|\) for every \(x\in X\setminus\{0\}\). A Banach space \(X\) is said to have the uniform Opial property if for every \(c>0\) there exists \(r=r(c)>0\) such that \(1+r\leq\liminf_{n\to\infty}\|x_n+x\|\) for every \(x\in X\) with \(\|x\|\geq c\) and every weakly null sequence \((x_n)\) in \(X\) with \(\liminf_{n\to\infty}\|x_n\|\geq 1\). A constant \(r_X(c)\) called Opial's modulus and defined for any Banach space \(X\) as follows: \(r_X(c)=\inf{\{\liminf_{n\to\infty}\|x_n+x\|-1\}}\), where \(c>0\) and the infimum is taken over all \(x\in X\) with \(\|x\|\geq c\) and all weakly null sequences \((x_n)\) in \(X\) such that \(\liminf_{n\to\infty}\|x_n\|\geq 1\). NEWLINENEWLINENEWLINELet \(\mathcal N\) and \(\mathcal R\) be the sets of natural and real numbers, respectively. \(l^0\) stands for the space of all real sequences. Denote the \(k\)-th coordinate term of a sequence \(x\in l^0\) by \(x(k)\) and the basic sequence by \(e_k=\chi_{\{k\}}\) \((k=1,2,\ldots)\), where \(\chi_A\) denotes the characteristic function of \(A\). The set \(\text{supp } x=\{i\in{\mathcal N}: x(i)\not=0\}\) is called the support of \(X\). A Banach space \((X,\|\cdot\|)\) is said to be a Köthe sequence space if \(X\) is a subspace of \(\ell^0\) such that NEWLINENEWLINENEWLINE(i) \(|x(i)|\leq|y(i)|\) for all \(i\in{\mathcal N}\), \(x\in \ell^0\) and \(y\in X\) imply \(x\in X\) and \(\|x\|\leq\|y\|\); NEWLINENEWLINENEWLINE(ii) there is \(x=(x(i))\in X\) such that \(\text{supp } x=\mathcal N\). NEWLINENEWLINENEWLINEWe say that an element \(x\) of a Köthe sequence space \(X\) is order continuous if for any sequence \((x_n)\) in \(X\) such that \(0\swarrow x_n(i)\leq|x_n(i)|\) for each \(i\in\mathcal N\) there holds \(\|x_n\|\to 0\). The set of all order continuous elements in \(X\) is denoted by \(X_a\). A Köthe sequence space \(X\) is said to be order continuous if \(X_a=X\). We say that a Köthe sequence space \(X\) have the Fatou property if for every \(y\in X\) and any sequence \((x_n)\in X\) satisfying \(|x_n(i)|\uparrow|y(i)\) for all \(i\in\mathcal N\) we have \(\|x_n\|\to\|y\|\). For \(1<p<\infty\), the Cesàro sequence space \(\text{ces}_p\) is defined by NEWLINE\[NEWLINE\text{ces}_p=\left\{x\inf^0:\|x\|=\left(\sum_{n=1}^\infty\left({1/n}\sum_{i= 1}^n|x(i)\right)^p \right)^{1/p}\right\}.NEWLINE\]NEWLINE The main results of the paper are the following theorems. NEWLINENEWLINENEWLINETheorem 1. Let \(X\) be a Köthe sequence space with the Fatou property. If \(r_x(1)>0\), then \(X\) is order continuous.NEWLINENEWLINENEWLINETheorem 2. If \(X\) is an order continuous Köthe sequence space, then for any \(c>0\) the Opial's modulus \(r_X(c)=\inf\{\liminf_{n\to\infty}\|x_n+x\|\}-1\), where the infimum is taken over all \(x\in X\) with \(\|x\|=c\) and all weakly null sequences \((x_n)\subset S(X)\) such that \(x=\sum_{i=1}^{i_1}x(i)e_i\) and \(x_n=\sum_{i=i_n+1}^{i_{n+1}}x_n(i)e_i\), \(n=1,2,\ldots,\) with \((i_n)\) being an increasing sequence of positive integers. NEWLINENEWLINENEWLINESome corollaries of these results are given. Let us mention some of them. NEWLINENEWLINENEWLINECorollary 1. Let \(\ell_p\) \((1<p<\infty)\) be a Lebesgue sequence space. Then \(r_{\ell_p}(c)= (1+c^p)^{1/p}-1.\) NEWLINENEWLINENEWLINECorollary 2. The space \(\text{ces}_p\) \((1<p<\infty)\) satisfies the uniform Opial's condition with \(r_{\text{ces}_p}(c)= (1+c^p)^{1/p}-1\).