Let $X = l^p$ and define $$A : X \to X$$ as $$ A(\mathbf{x}) = (x_1, x_2/2, x_3/3,\dots),\ \ \mathbf{x}=(x_1,x_2,x_3,\dots).$$ Find spectrum, eigenspectrum and approximate eigenspectrum for $A$.
Spetrum:$$ \sigma(A)=\{k\in \mathbb{K}\ :\ A-kI\ \text{is not invertible}\} $$
Eigen Spetrum:$$ \sigma_e(A)=\{k\in \mathbb{K}\ :\ A-kI\ \text{is not injective}\} $$
Approximate Spetrum:$$ \sigma_a(A)=\{k\in \mathbb{K}\ :\ A-kI\ \text{is not bounded below}\} $$
Here, in this question $$(A-kI)(\mathbf{x})=\left(x_1(1-k),x_2\left(\frac{1}{2}-k\right),x_3\left(\frac{1}{3}-k\right),\dots\right).$$ Now how will I find out $k$ for the all spectrum?