Example of two non-zero operators $A$ and $B$ such that $AB=0$

Question

Let $E$ be a complex Hilbert space.

I look for an example of $A,B\in \mathcal{L}(E)$ such that $A\neq 0$ and $B\neq 0$ but $AB=0$.

Answer 1

Let $A=E_{1,1}$ and $B=E_{2,2}$, then $AB=0$, where $E_{1,1}$ is the matrix has scalar $1$ only at $(1,1)$ entry, $E_{2,2}$ is the matrix has scalar $1$ only at $(2,2)$ entry.

Answer 2

In $\ell^2(\mathbb{C})$, define$$A(x_1,x_2,x_3,\ldots)=(x_1,0,x_3,0,x_5,0,\ldots)\text{ and }B(x_1,x_2,x_3,\ldots)=(0,x_2,0,x_4,0,x_6,\ldots).$$