On generalized \(\ell_p\)-spaces (Q2643497)

For \(n\in\mathbb{N}\) and \(p\geq 1\), let \(\ell_p(n)\) denote the space of \(\overline x= (x_1, x_2,\dots, x_n)\in\mathbb{C}^n\) endowed with the \(\|\cdot\|^n_p\) norm \[ \|\overline x\|^n_p= \begin{cases} (\sum^n_{k=1} |x_k|^p)^{1/p},& 1\leq p<\infty,\\ \max_{1\leq k\leq n}|x_k|, & p= \infty.\end{cases} \] The above norm has the property \(\|\overline x\|^n_p= \|\,|\overline x|\,\|^n_p\), \(|\overline x|= (|x_1|,|x_2|,\dots, |x_n|)\). This property characterizes \(\|\cdot\|^n_p\) as an absolute norm. In general, a norm \(\|\cdot\|\) on \(\mathbb{C}^n\) is said to be absolute if \(\|\overline x\|= \|\,|\overline x|\,\|\) for every \(\overline x= (x_1,x_2,\dots, x_n)\in\mathbb{C}^n\). Moreover, a norm \(\|\cdot\|\) on \(\mathbb{C}^n\) will be called normalized if \(\| e^{(k)}\|= 1\) for the canonical basis \(\{e^{(k)},\;k= 1,2,\dots, n\}\) of \(\mathbb{C}^n\). In [J.~Math.\ Anal.\ Appl.\ 252, No.\,2, 879--905 (2000; Zbl 0999.46008)], the second of the present authors together with \textit{M. Kato} and \textit{Y. Takahashi}, generalizing a result by F. F. Bonsal and J. Dunkan, showed that the family \(AN_n\) of all absolute normalized norms on \(\mathbb{C}^n\) is in one-to-one correspondence, under the equations \[ \psi(\overline s)= \Biggl\|\Biggl(1- \sum^{n-1}_{i=1} s_i,s_1,\dots, s_{n-1}\Biggr)\Biggr\|,\quad \|\cdot\|\in AN_n, \] \[ \|\overline s\|_\psi= \begin{cases} Q\psi\left({|s_2|\over Q},\dots, {|s_n|\over Q}\right), & s\neq \theta,\\ 0, & s=\theta,\end{cases}\quad \psi\in\Psi_n, \] (\(s\in \mathbb{C}^n\), \(\theta= (0,0,\dots, 0)\), \(Q= \sum^n_{i=1} |s_i|\)), with the set \(\Psi_n\) of all continuous convex functions satisfying certain given conditions on the set \[ \Delta_n= \Biggl\{s= (s_1,\dots, s_{n-1})\in \mathbb{R}^{n-1}: \sum^{n-1}_{i=1} s_i\leq 1,\, s_i\geq 0\Biggr\}. \] In the present paper, the authors extend the above result to certain spaces of the \(\ell_p\) type. They start by defining a norm on the space \(\ell_0\) of all infinite sequences of complex numbers with finitely many nonzero elements to be absolute if \(\|\overline x\|=\|\,|\overline x|\,\|\) for every \(\overline x= (x_1, x_2,\dots)\in \ell_0\), \((|\overline x|= (|x_1|,|x_2|,\dots))\) and normalized if \(\| e^{(n)}\|= 1\), \(n\in\mathbb{N}\), \(e^{(n)}= (0,0,\dots, 0,1,0,\dots)\) with \(1\) at the \(n\)-th position. The family of all absolute normalized norms on \(\ell_0\) is denoted by \(AN_\infty\). Continuing, they consider the set \(\Psi_\infty\) of all continuous convex functions satisfying certain given conditions on the set \[ \Delta_\infty= \Biggl\{s= (s_n)_{n\in\mathbb{N}}\in \ell_0: \sum^\infty_{n=1} s_n= 1,\;s_n\geq 0\;(\forall n)\Biggr\} \] and they prove that the sets \(AN_\infty\) and \(\Psi_\infty\) are in one-to-one correspondence under the equation \(\psi(s)=\| s\|\). Finally, using the functions in \(\Psi_\infty\), the authors introduce a notion of generalized \(\ell_p\)-spaces, the \(\ell_\psi\)-spaces, by associating, to each function \(\psi\in \Psi_\infty\), the respective unique norm \(\|\cdot\|_\psi\in AN_\infty\) and the space \[ \ell_\psi= \Bigl\{x= (x_n)_{n\in\mathbb{N}}\in \ell_\infty: \lim_{n\to\infty}\|(x_1, x_2,\dots, x_n,0, 0,\dots)\|_\psi< \infty\Bigr\} \] (\(\ell_\infty\) is the space of all bounded complex sequences) and study their structure.