Some Remarks on the Matrix Domain and the Spectra of a Generalized Difference Operator

Abstract

Recently, Meng and Mei (J Math Anal Appl, 2018. https://doi.org/10.1016/j.jmaa.2018.10.051) have introduced two Euler difference sequence spaces $$e_0^t(B_\nu ^{(m)})$$ and $$e_c^t(B_\nu ^{(m)})$$ using generalized difference operator $$B_\nu ^{(m)}$$. Some functional properties and Kothe–Toeplitz duals of the proposed spaces have been determined. Also, the fine spectra of the generalized difference operator $$B_\nu ^{(m)}$$ over the sequence space $$\ell _1$$ have been computed following the results of Dutta and Baliarsingh (Appl Math Comput 219:1776–1784, 2012). In fact, the results for the spectra of the said difference operators are new and interesting. Upon further investigations, it has been noticed that these results should be improved and modified. The primary objective of this note is to provide such sorts of substantial modifications on these results.