Unitary equivalence of an operator with a double copy of itself?

Question

I have been stacked with the following question: How can one finds an $A\in B(H)$ (non trivial) which is unitarily equivalent to $A\oplus A$?

Thanks in advance.

Math.

Answer 1

Let $\{e_n\}_{n\in\mathbb N}$ be an orthonormal basis of $H$, and let $P$ be the projection onto the closed linear span of $\{e_{2n}\}_{n\in\mathbb N}$. Let $U:H\to H\oplus H$ be the operator defined by linear extension of the map taking $e_1$ to $(e_1,0)$, $e_2$ to $(e_2,0)$, $e_3$ to $(0,e_1)$, $e_4$ to $(0,e_2)$, and so on. Then $U$ is a unitary, and $P\oplus P=UPU^*$.