Let $A \in M_n(\mathbb{C})$ be nilpotent matrix such that $A^{4} = 0$, We can find matrix $B \in M_n(\mathbb{C})$ such that we have $$AB \neq 0 \ \ and \ \ BA = 0$$ The question is as follows:
How can we extend this property to the infinite dimensionl? if A is nilpotent operators such that $A^{4} = 0$ in infinite dimentional banach space.
Suppose $A$ is an operator over a vector space $V$ such that $A^k =0$ for some finite $k > 1$ and that $A^{k-1} \neq 0$. There must exist a vector $v \in V$ such that $A^{k-1} v \neq 0$. Moreover, there exists a linear functional $f:V \to \Bbb F$ such that $f(A^{k-1} v) \neq 0$.
Define $B$ to be the operator $$ B(x) = f(A^{k-1}x)v. $$ Verify that $B \circ A = 0$. On the other hand, $A \circ B$ cannot be zero because $$ A(Bv) = A(f(A^{k-1}v)v) = f(A^{k-1}v)Av \neq 0. $$