Some Spectrum Estimates of the $\alpha q$-Ces\`{a}ro Matrices with $0<\alpha ,q<1$ on $c$

Abstract

$q$-Calculus Theory is rapidly growing in various directions. The goal of this paper is to collect and underline recent results on $\alpha q$-analogs of the Cesàro matrix andemphasize various generalizations. One $\alpha q$-analogs of the Cesàro matrix of order one is the triangular matrix with nonzero entries $c_{nk}^{\alpha }\left( q\right) =\tfrac{\left( \alpha q\right) ^{n-k}}{1+q+\cdots +q^{n}},\ 0\leq k\leq n$, where $\alpha ,q\in \left( 0,1\right) $. The purpose of this article examines various spectral decompositions of $C_{q}^{\alpha }=\left( c_{nk}^{\alpha }\left( q\right) \right) $ such as the spectrum, the fine spectrum, the approximate point spectrum, the defect spectrum, and the compression spectrum on the sequence space $c$.