On the fine spectra of triangular Toeplitz operators Muhammed Altun Faculty of Arts and Sciences, Adıyaman University, 02040 Adıyaman, Turkey article info Keywords: Spectrum of an operator Toeplitz operators Banded matrices abstract The fine spectra of triangular double-band and triple-band matrices were examined by sev- eral authors. Here we determine the fine spectra of Toeplitz operators, which are repre- sented by upper and lower triangular n-band infinite matrices, over the sequence spaces c 0 and c. Also some spectral results over ‘ 1 are given. Ó 2011 Elsevier Inc. All rights reserved. 1. Introduction In functional analysis, the spectrum of an operator is a generalization of the notion of eigenvalues for matrices. By fine spectrum of an operator over a Banach space we mean the calculation of the three parts of the spectrum, which are the point spectrum, the continuous spectrum and the residual spectrum. A general Toeplitz operator is represented by a banded matrix as T ¼ a 0 a 1 a 2 a 3 a 4  a 1 a 0 a 1 a 2 a 3  a 2 a 1 a 0 a 1 a 2  a 3 a 2 a 1 a 0 a 1  a 4 a 3 a 2 a 1 a 0  . . . . . . . . . . . . . . . . . . 2 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 5 : We will focus on two kinds of Toeplitz operators. The first one is represented by a lower triangular n-band matrix, so it has the form T ½L ðaÞ¼ a 0 0 0 0 0  a 1 a 0 0 0 0  a 2 a 1 a 0 0 0  . . . . . . . . . . . . . . . . . . a n1 a n2 a n3  a 0  0 a n1 a n2 a n3   0 0 a n1 a n2 a n3  . . . . . . . . . . . . . . . . . . 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 ; where a ¼ða 0 ; a 1 ; ... ; a n1 Þ is a complex n-tuple with a n1 – 0. And the second kind is represented by an upper triangular n- band matrix, which is the transpose of a lower triangular n-band matrix. Let 0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.03.003 E-mail addresses: maltun@adiyaman.edu.tr, muhammedaltun@gmail.com Applied Mathematics and Computation 217 (2011) 8044–8051 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc