Let $X$ be Banach space over a field $\mathbb{C}$. Consider the Banach space $L_c$ of compact operators in $X$. Let $A^0\in L_c$ be fixed and $\lambda^0\neq 0$ his eigenvalue with algebraic multiplicity $m$ which is associated with Jordan form $J_{\overline k}(\lambda^0)$.

In the canonical basis of the subspace that corresponding to eigenvalue $\lambda^0$, operator $A^0$ has the form $$ A^0=\begin{pmatrix}J_{\overline k}(\lambda^0)&0 \\\ 0 & A_{\textbf{2,2}}^0 \end{pmatrix}. $$

Questions.

1. Can I speak about canonical basis in Banach space or better use a Hilbert space?
2. Is it natural to consider the Jordan form of compact operator on Banach or Hilbert space?
3. Does any non-zero eigenvalue of a compact operator have finite multiplicity?

21.11. Operator $A^0$ such that it has non-zero eigenvalues.