Pair of normal operators

Question

Let $\mathcal{B}(F)$ the algebra of all bounded linear operators on a complex Hilbert space $F$.

$S=(S_1,S_2)$ is called normal iff $S_1S_2=S_2S_1$ and both $S_1$ and $S_2$ are normal.

Assume that $F$ is an infinite-dimensional complex Hilbert space.

I look for an example of two normal operators $S_1$ and $S_2$ (which are not a scalar multiple of the identity) such that $S_1S_2=S_2S_1$ and $S_1\neq S_2$.

Answer 1

Here's a family of examples that you might find interesting. Take any two bounded sequences $(a_n),(b_n)$. Define the maps $S_i:\ell_2 \to \ell_2$ by $$ (T_1x)_n = a_nx_n, \qquad (T_2x)_n = b_n x_n $$