In this paper, we investigate the necessary conditions on any s-type sequence space to form an operator ideal. As a result, we show that the s-type Nakano generalized difference sequence space X fails to generate an operator ideal. We investigate the sufficient conditions on X to be premodular Banach special space of sequences and the constructed prequasi-operator ideal becomes a small, simple, and closed Banach space and has eigenvalues identical with its s-numbers. Finally, we introduce necessary and sufficient conditions on X explaining some topological and geometrical structures of the multiplication operator defined on X.

In this article, we investigate the notion of the pre-quasi norm on a generalized Cesàro backward difference sequence space of non-absolute type $(\Xi (\Delta,r) )_{\psi }$( Ξ ( Δ , r ) ) ψ under definite function ψ. We introduce the sufficient set-up on it to form a pre-quasi Banach and a closed special space of sequences (sss), the actuality of a fixed point of a Kannan pre-quasi norm contraction mapping on $(\Xi (\Delta,r) )_{\psi }$( Ξ ( Δ , r ) ) ψ , it supports the property (R) and has the pre-quasi normal structure property. The existence of a fixed point of the Kannan pre-quasi norm nonexpansive mapping on $(\Xi (\Delta,r) )_{\psi }$( Ξ ( Δ , r ) ) ψ and the Kannan pre-quasi norm contraction mapping in the pre-quasi Banach operator ideal constructed by $(\Xi (\Delta,r) )_{\psi }$( Ξ ( Δ , r ) ) ψ and s-numbers has been determined. Finally, we support our results by some applications to the existence of solutions of summable equations and illustrative examples.

We suggest a sufficient setting on any linear space of sequences $\mathcal{V}$V such that the class $\mathbb{B}^{s}_{\mathcal{V}}$BVs of all bounded linear mappings between two arbitrary Banach spaces with the sequence ofs-numbers in $\mathcal{V}$V constructs a map ideal. We define a new sequence space $(\mathit{ces}_{r_{1},r_{2}}^{t} )_{\upsilon }$(cesr1,r2t)υ for definite functionalυby the domain of $(r_{1},r_{2})$(r1,r2) -Cesàro matrix in $\ell _{t}$ℓt , where $r_{1},r_{2}\in (0,\infty )$r1,r2∈(0,∞) and $1\leq t<\infty $1≤t<∞ . We examine some geometric and topological properties of the multiplication mappings on $(\mathit{ces}_{r_{1},r_{2}}^{t} )_{\upsilon }$(cesr1,r2t)υ and the pre-quasi ideal $\mathbb{B}^{s}_{ (\mathit{ces}_{r_{1},r_{2}}^{t} )_{\upsilon }}$B(cesr1,r2t)υs .