Equivalent conditions for reducibility of $\mathbb{C}$-linear operator on ${\mathbb{C}}^n$

Question

Let $A$ be a $\mathbb{C}$-linear operator from ${\mathbb{C}}^n$ to itself, with rank$A=p < n$.

For complex linear subspace $E$ of ${\mathbb{C}}^n$, $A$ is called reduced by $E$ if both $E$ and $E^{\bot}$ are invariant subspaces of $A$.

Note that $A$ induces the orthogonal decompositions

$$ {\mathbb{C}}^n = \text{Ran}A\oplus\text{Ker}A^* =\text{Ran}A^*\oplus\text{Ker}A. $$

Then we claim the following characterization of reducibility of $A$ by Ran$A$ :

Proposition) The following conditions are equivalent

i) Ran$A$ reduces $A$

ii) Ran$A$ = Ran$A^* $, and Ker$A$ = Ker$A^*$

iii) $A$ commutes with the orthogonal projection onto Ran$A$

iv) there exists a unitary operator $U : {\mathbb{C}}^n \rightarrow {\mathbb{C}}^n$ and a $\mathbb{C}$-linear isomorphism $B : {\mathbb{C}}^p \rightarrow {\mathbb{C}}^p$ such that, for all $(z,w)\in{\mathbb{C}}^p\times{\mathbb{C}}^{n-p},$ $$ \left(B(z),0\right) = UAU^*(z,w).$$

How can I prove this?

Note: this is lemma 3.1.10 from "Pluripotential Theory" by Klimek.

Answer 1

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I hope this answers your question.