Fine Spectra of Upper Triangular Double - Band Matrices over the Sequence Space p , 1 < p < ∞

The operator A r̃, s̃ on sequence space on p is defined A r̃, s̃ x rkxk skxk 1 ∞ k 0, where x xk ∈ p, and r̃ and s̃ are two convergent sequences of nonzero real numbers satisfying certain conditions, where 1 < p < ∞ . The main purpose of this paper is to determine the fine spectrum with respect to the Goldberg’s classification of the operator A r̃, s̃ defined by a double sequential band matrix over the sequence space p. Additionally, we give the approximate point spectrum, defect spectrum, and compression spectrum of the matrix operator A r̃, s̃ over the space p.