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

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<\infty $ 1 ≤ t < ∞ . 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 .