Certain domains of a new matrix constructed by Euler totient and its summation function

With the aid of the Euler totient function $\phi$ and its summation function $\mathrm{\top }$, a new matrix $\mathrm{\Delta }(\phi ,\mathrm{\top })=(\delta (\phi ,\mathrm{\top }{)}_{nk})$, where

$$\delta (\phi ,\mathrm{\top }{)}_{nk}=\{\begin{array}{cc}\frac{(-1{)}^{n-k}\mathrm{\top }(k)}{\phi (n)},& n-1\le k\le n,\\ 0,& \text{otherwise}\end{array}$$

is constructed to define the domains ${\ell }_p(\mathrm{\Delta }(\phi ,\mathrm{\top }))$, ${\ell }_{\mathrm{\infty }}(\mathrm{\Delta }(\phi ,\mathrm{\top }))$, and ${\ell }_1(\mathrm{\Delta }(\phi ,\mathrm{\top }))$. After obtaining the norms on these domains, it is proved that these spaces are linearly isomorphic to classical ones. Also, their dual spaces are determined. Finally, characterizations of several matrix mappings are stated and proved.