Let $T$ be a bounded linear operator on a Hilbert space $H$. Define what it means to say that $T$ is (i) compact, and (ii) Fredholm. What is the index, ind $(T)$, of a Fredholm operator $T$ ?

Let $S, T$ be bounded linear operators on $H$. Prove that $S$ and $T$ are Fredholm if and only if both $S T$ and $T S$ are Fredholm. Prove also that if $S$ is invertible and $T$ is Fredholm then $\operatorname{ind}(S T)=\operatorname{ind}(T S)=\operatorname{ind}(T)$.

Let $K$ be a compact linear operator on $H$. Prove that $I+K$ is Fredholm with index zero.