examples of a compact operator $K: l^2 \to l^2$ with point spectrum $\sigma_p(K)=\{0,1,\frac{1}{2},\frac{1}{3},...\}$, with finite and infinite kernel

Question

I want to give examples of a compact operator $K: l^2 \to l^2$ with point spectrum $\sigma_p(K)=\{0,1,\frac{1}{2},\frac{1}{3},...\}$

(a) finite-dimensional $Ker(K)$,

(b) infinite-dimensional $Ker(K)$.

My attempt

a) - let $K(x_1,x_2,x_3,...)=(0,x_1,\frac{x_2}{2},\frac{x_3}{3},...) $ , then $ker(K)=\{0\}$

• let $K(x_1,x_2,x_3,...)=(0,x_2,\frac{x_3}{2},\frac{x_4}{3},...) $ , then $ker(K)=\{x\in l^2 : x = (x_1,0,0,...), \ \ x_1 \in \mathbb{R}\}$

b) - let $K(x_1,x_2,x_3,...)=(0,x_2,0,\frac{x_4}{2},0,\frac{x_6}{3},...) $ , then $ker(K)=\{x\in l^2 : x = (x_1,0,x_3,0,x_5,...), \ \ x_{2k-1} \in \mathbb{R} , \forall k\ge 1\}$

Answer 1

For (a) define $K(x_1,x_2,...)=\sum x_i a_ie_i$ where $(e_i)$ is the usual orthonormal basis and $(a_n)$ is the given sequence $(0,1,\frac 1 2 ,...)$.

For (b) define $K(x_1,x_2,...)=\sum_{i \, \text {even}} x_{i} a_ie_i$. I will leave the verifications to you.