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We suggest a sufficient setting on any linear space of sequences    $\mathcal{V}$  V such that the class    $\mathbb{B}^{s}_{\mathcal{V}}$  B V s of all bounded linear mappings between two arbitrary Banach spaces with the sequence of  s -numbers in    $\mathcal{V}$  V constructs a map ideal. We define a new sequence space    $(\mathit{ces}_{r_{1},r_{2}}^{t} )_{\upsilon }$  ( ces r 1 , r 2 t ) υ for definite functional  υ  by the domain of    $(r_{1},r_{2})$  ( r 1 , r 2 ) -Cesàro matrix in    $\ell _{t}$  ℓ t , where    $r_{1},r_{2}\in (0,\infty )$  r 1 , r 2 ∈ ( 0 , ∞ ) and    $1\leq t&lt;\infty $  1 ≤ t &lt; ∞ . We examine some geometric and topological properties of the multiplication mappings on    $(\mathit{ces}_{r_{1},r_{2}}^{t} )_{\upsilon }$  ( ces r 1 , r 2 t ) υ and the pre-quasi ideal    $\mathbb{B}^{s}_{ (\mathit{ces}_{r_{1},r_{2}}^{t} )_{\upsilon }}$  B ( ces r 1 , r 2 t ) υ s .

$(r_{1},r_{2})$-Cesàro summable sequence space of non-absolute type and the involved pre-quasi ideal