On spaces derivable from a solid sequence space and a non-negative lower triangular matrix (Q2814821)

Let \(A=C_{1},\) the Cesàro matrix. For \(p>1,\) the Cesàro sequence space NEWLINE\[NEWLINE \mathrm{ces}_{p}:=\left\{ x=\left( x_{k}\right) :A\left| x\right| :=\left( \frac{1}{n}\sum_{k=1}^{n}\left| x_{k}\right| \right) _{n\in \mathbb{N}}\in\ell_{p}\right\} NEWLINE\]NEWLINE is a Banach lattice with the norm \(x\mapsto\left\| A\left| x\right| \right\| _{p}.\) Define the projective sequence \(\left( \Lambda_{k}\right) _{k=0,1,\dots},\) where \(\Lambda_{0}:=\ell_{p}\), \(\Lambda_{1}:=\mathrm{ces}_{p}\) and \(\Lambda_{k}:=\left\{ x=\left( x_{k}\right) :A\left| x\right| \in\Lambda_{k-1}\right\} \) for \(k>1\). Then \(\Lambda_{k-1}\subset\Lambda_{k}\) and the Banach lattice \(\Lambda_{k}\) has the sectional convergence property.NEWLINENEWLINEThe authors define \(X:=\left\{ \left( x^{\left( k\right) }\right) \in\prod\limits_{k=0}^{\infty}\Lambda_{k}:Ax^{\left( k\right) }=x^{\left( k-1\right) }\text{ }\left( k\in\mathbb{N}\right) \right\} \), with respect to the product topology \(X\) is a Fréchet space. If \(P_{0},P_{1},\dots\) are the projections, then NEWLINE\[NEWLINEX=\left\{ \left( x,A^{-1}x,\left( A^{-1}\right) ^{2}x,\dots\right) :x\in P_{0}\left( X\right) \right\} NEWLINE\]NEWLINE and \(c_{00}\subset P_{0}\left( X\right) \subset\ell_{p}\) (here, \(c_{00}\) is the space of eventually zero sequences). The authors show that (1) \(P_{0}\left( X\right) \) is not solid and also not closed in \(\left( \ell_{p},\left\| \;\right\| _{p}\right) ,\) (2) the onto map \(T:\left( P_{0}\left( X\right) ,\left\| \;\right\| _{p}\right) \rightarrow\left( P_{1}\left( X\right) ,\left\| \;\right\| _{\mathrm{ces}_{p}}\right) ,\) \ \(x\mapsto A^{-1}x\) is not continuous, and (3) the topology on \(X\) defined by the norm \(\left( x^{\left( k\right) }\right) \mapsto\left\| x^{\left( 0\right) }\right\| _{p}\) does not coincide with the topology induced by the product topology.