# Cesaro and Tandori sequence spaces, representations and duality

Definitions. Fix $1\leq p\leq\infty$. Given a scalar sequence $a=(a_n)_{n=1}^\infty$, denote by $\tilde{a}=(\tilde{a}_n)_{n=1}^\infty$, where each $\tilde{a}_n=\sup_{k\geq n}|a_k|$. Now we define the $\boldsymbol{p}$-Tandori sequence space as the vector space $\widetilde{\ell}_p$ of scalar sequences $a$ such that \begin{equation}\|a\|_{\widetilde{p}}:=\|\tilde{a}\|_p<\infty,\end{equation} where $\|\cdot\|_p$ denotes the usual $\ell_p$ norm.

We define the $\boldsymbol{p}$-Cesàro sequence space as follows. For a scalar sequence $a=(a_n)_{n=1}^\infty$ let us define $c(a)=\left(\frac{1}{n}\sum_{k=1}^n|a_k|\right)_{n=1}^\infty$, and denote by $ces_p$ the space of all scalar sequences $a$ such that \begin{equation}\|a\|_{c(p)}:=\|c(a)\|_p<\infty.\end{equation}

Note that the canonical unit vectors $(e_n)_{n=1}^\infty$ form a (nonnormalized) basic sequence in $\widetilde{\ell}_p$, and this is a basis for the space as long as $p\neq\infty$. The same holds for $ces_p$. Note also that these spaces are all complete, i.e. all Banach spaces.

Background. The space $ces_p$ is reflexive for $1<p<\infty$. In Corollary 3 of this paper it is shown that \begin{equation}\overline{\ell}_1\cong\left(\oplus_{n=0}^\infty\ell_\infty^{2^n}\right)_1\end{equation} and \begin{equation}ces_\infty=\overline{\ell}_1^*=\left(\oplus_{n=0}^\infty\ell_1^{2^n}\right)_\infty.\end{equation} It is also known that $ces_\infty^0$ (the closed span $[e_n]_{n=1}^\infty$ in $ces_\infty$) that \begin{equation}ces_\infty^{0*}=\widetilde{\ell}_1.\end{equation}

Question 1. Is it true that $ces_\infty^0\cong\left(\oplus_{n=0}^\infty\ell_1^{2^n}\right)_{c_0}?$

Question 2. Is it true that $ces_p^*\cong\widetilde{\ell}_{p'}$, $\frac{1}{p}+\frac{1}{p'}=1$, for $1<p<\infty$?

Question 3. Do $ces_p$ and $\widetilde{\ell}_p$ have similar representationss as above, i.e. something like $\left(\oplus_{n=0}^\infty\ell_{p'}^{2^n}\right)_p$ or something like that?

Question 4. What is the dual and/or predual of $ces_1$?

Thank you!