How to prove the resolvent of this operator is compact? [closed]
Solution 1:
We can approximate $R_\lambda(A)$ by finite-rank operators with respect to the operator norm by defining $$ R_\lambda^{(k)}(A)(v)=\left(\frac{v_1}{1-\lambda},\dots,\frac{v_k}{k-\lambda},0,0,\dots\right). $$