I am having some trouble proving this: Let $T_1\in H_1$ and $T_2\in H_2$ where $H_1,H_2$ are Hilbert spaces. Let $T=T_1\oplus T_2$ on $H_1\oplus H_2$. I need to show $T$ is compact iff $T_1$ and $T_2$ are compact.
I was able to prove it in the reverse direction, but I am having trouble proving that if T is compact then $T_1$ and $T_2$ are compact.