# Pair of normal operators

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 $S_1\neq S_2$.

• How about $S_1 = I$ and $S_2 = 2I$? – Omnomnomnom Apr 4 '18 at 10:47
• @Omnomnomnom You are right but I hope to get a non trivial example and thank you for your help – Schüler Apr 4 '18 at 10:49
• My point is that you should be clear about what "non-trivial" means in this context – Omnomnomnom Apr 4 '18 at 10:50
• Notably, every finite dimensional example can be thought of (up to a change of basis) as a pair of diagonal matrices. Would versions of these examples be considered "trivial"? – Omnomnomnom Apr 4 '18 at 10:51
• @Omnomnomnom Please see my edit. Thank you. – Schüler Apr 4 '18 at 10:52

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 $$(T_1x)_n = a_nx_n, \qquad (T_2x)_n = b_n x_n$$