Spectrum and fine spectrum of generalized lower triangular triple band matrices over the sequence space l_p

Abstract

The spectrum of triangular band matrices defined on the sequence spaces where the entries of each band is a constant or convergent sequence is well studied. In this article, the spectrum and fine spectrum of a new generalised difference operator defined by a lower triangular triple band matrix on the sequence space lp(1 ≤ p < ∞) are obtained where the bands are considered as periodic sequences. The approximate point spectrum, defect spectrum, compression spectrum and the Goldberg classification of the spectrum are also discussed. Suitable examples are given in order to supplement the results. Several special cases of our findings are discussed which confirm that our study is more general and extensive.

Spectral Properties of Jacobi-like Band Matrices on the Sequence Space $\ell_p$

“…The spectral analysis of operators defined over sequence spaces has been treated by many researchers worldwide. The literature contains the spectrum and fine spectrum of several classes of Toeplitz operators [2,6,9], Cesàro operators [1,7,14], Rhaly operators [21,22], operators generated by various difference equations [10,11,17,18,19], etc. For a detailed review, one may refer to the survey articles [4,20] and the references therein.…”