Matrix Transformations and Generators of Analytic Semigroups

In this paper, we are interested in the study of operators represented by infinite matrices. Note that in [1], Altay and Başar gave some results on the fine spectrum of the difference operator Δ acting on the sequence spaces c0 and c. Then they dealt with the fine spectrum of the operator B(r,s) defined by a matrix band over the sequence spaces c0 and c. In de Malafosse [3, 5], there are results on the spectrum of the Cesàro matrix C1 and on the matrix Δ considered as operators from sr to itself. Spectral properties of unbounded operators are used in the theory of the sum of operators. The notion of generators of analytic semigroup was developed in this way. Recall that this theory was studied by many authors such as Da Prato and Grisvard [2, 12], Fuhrman [11], Labbas and Terreni [16, 17]. Some applications can also be found in Labbas and de Malafosse [15] of the sum of operators in the theory of summability in the noncommutative case. Some results were obtained in de Malafosse [4] on the equation