Thai Journal of Mathematics Volume x (xxxx) Number x : xx–xx www.math.science.cmu.ac.th/thaijournal Online ISSN 1686-0209 Notes on the Spectrum of Lower Triangular Double-Band Matrices Ali M. Akhmedov and Saad R. El-Shabrawy Faculty of Mechanics and Mathematics, Baku State University Z. Khalilov Str., 23, AZ 1148, Baku, Azerbaijan e-mail : akhmedovali@rambler.ru (A.M. Akhmedov), srshabrawy@yahoo.com (S.R. El-Shabrawy) Abstract : In the paper by Srivastava and Kumar [P.D. Srivastava, S. Kumar, Thai J. Math. 8 (2) (2010) 221–233], the authors have introduced the lower triangular double-band matrix Δ v as an operator on the sequence space l 1 and studied the spectrum and fine spectrum of this operator over l 1 . The operator Δ v on l 1 is defined by Δ v x =(v k x k − v k-1 x k-1 ) ∞ k=0 with x -1 =0, where x =(x k ) ∈ l 1 and (v k ) is either constant or strictly decreasing sequence of positive real numbers satisfying certain conditions. In this paper we give notes on the point spectrum and the residual spectrum of the operator Δ v over the space l 1 in the case when (v k ) is a strictly decreasing sequence of positive real numbers. Keywords : Spectrum of an operator; Generalized difference operator; Sequence spaces. 2010 Mathematics Subject Classification : 47A10; 47B37. 1 Introduction, Preliminaries and Notation Let X and Y be Banach spaces and T : X → Y be a bounded linear operator. By R(T ), we denote the range of T , i.e., R(T )= {y ∈ Y : y = Tx, x ∈ X } . By B(X ), we denote the set of all bounded linear operators on X into itself. If T ∈ B(X ), then the adjoint T * of T is a bounded linear operator on the dual X * Copyright c  2011 by the Mathematical Association of Thailand. All rights reserved.