Mediterr. J. Math. (2020) 17:121 https://doi.org/10.1007/s00009-020-01557-9 c  Springer Nature Switzerland AG 2020 Ces` aro Spaces and Norm of Operators on These Matrix Domains H. Roopaei, D. Foroutannia, M. ˙ Ilkhan and E. E. Kara Dedicated to Prof. Grahame Bennett for his brilliant works in sequence spaces. Abstract. In this paper, we investigate some properties of the domain of the Ces`aro matrix of order n and compute K¨othe dual of this space. Moreover, we compute the norm of well-known operators such as Hilbert, Hausdorff, backward difference and weighted mean operators on this new sequence space. Mathematics Subject Classification. 26D15, 40C05, 40G05, 47B37, 47B39. Keywords. Matrix operator, norm, Ces`aro matrix, Hausdorff matrix, Hilbert matrix, backward difference operator, sequence space. 1. Introduction Let ω denote the set of all real-valued sequences. Any linear subspace of ω is called a sequence space. The Banach space ℓ p is the set of all real sequences x =(x k ) ∞ k=0 ∈ ω such that ‖x‖ p =  ∞  k=0 |x k | p  1/p < ∞ (1 ≤ p< ∞). By c and ℓ ∞ , we denote the spaces of all convergent and bounded real se- quences, respectively. These spaces are Banach spaces with the norm ‖x‖ ∞ = sup k |x k |. Here and in the rest of the paper, the supremum is taken over all k ∈ N 0 = {0, 1, 2, 3,...}. Also, we use the notion N = {1, 2, 3,...}. One can consider an infinite matrix as a linear operator from a sequence space to another one. Given any two arbitrary sequence spaces X, Y and an infinite matrix T =(t i,j ), we define a matrix transformation from X into Y as Tx = ((Tx) i )=  ∑ ∞ j=0 t i,j x j  provided that the series is convergent for each i ∈ N 0 . By (X, Y ), we denote the family of all infinite matrices from X into Y . 0123456789().: V,-vol