This work mainly explores the idea of the spectrum of the fractional difference operator ${\mathrm{\Delta }}_{\nu }^{\left(\alpha \right)}$ (of order $\alpha \in \mathbb{R}$) over the Banach space ℓ₁. The difference operator ${\mathrm{\Delta }}_{\nu }^{\left(\alpha \right)}$ via the sequence x = (x_k) is defined by $({\mathrm{\Delta }}_{\nu }^{\left(\alpha \right)}x{)}_k=\sum _{i=0}^{\mathrm{\infty }}\frac{(-\alpha {)}_i}{i!}{\nu }_{k-i}{x}_{k-i},$where (α)_i denotes the Pochhammer symbol, $\alpha \in \mathbb{R}$ and ν = (ν_k) is either a constant or strictly decreasing sequence of positive real numbers with some conditions. Sets comprising of the point, the residual and the continuous spectrum of the operator ${\mathrm{\Delta }}_{\nu }^{\left(\alpha \right)}$ on the Banach space ℓ₁ are computed.

中文翻译：

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关于分数差算子的谱

这项工作主要探讨分数差算子谱的思想${\mathrm{△}}_{\nu }^{\left(\alpha \right)}$（按顺序$\alpha \in \mathbb{R}$) 在 Banach 空间 ℓ ₁上。差分算子${\mathrm{△}}_{\nu }^{\left(\alpha \right)}$通过序列x  = ( x _k ) 定义为$({\mathrm{△}}_{\nu }^{\left(\alpha \right)}X{)}_k=\sum _{我=0}^{\mathrm{\infty }}\frac{(-\alpha {)}_{我}}{我！}{\nu }_{k-我}{X}_{k-我},$其中 ( α ) _i表示 Pochhammer 符号，$\alpha \in \mathbb{R}$并且ν  = ( ν _k ) 在某些条件下是常数或严格递减的正实数序列。由算子的点、残差和连续谱组成的集合${\mathrm{△}}_{\nu }^{\left(\alpha \right)}$在 Banach 空间 ℓ ₁上计算。