Properties of compact operators I'm working on the following problem, regarding compact operators.

Let $T$ and $S$ be bounded linear operators on an infinite-dimensional Hilbert space. Prove or disprove the following statements:

*

*If $T^n=I$ for some $n$, then $T$ is not compact.

*If $T^2=0$ then $T$ is compact.

*If $TS$ is compact then either $T$ or $S$ is compact.


The definition of compactness that I'm familiar with is: a linear operator $T$ is compact if and only if whenever $(x_n)$ is a bounded sequence, $(Tx_n)$ contains a convergent subsequence.
What I've done: For (1), I have seen the proof that an idempotent operator is compact if and only if it has finite rank. For (2), I am not sure how to start. For (3), I have only been able to prove the converse (if either $T$ or $S$ is compact than $TS$ is compact).
Any help on this problem? Thanks!

*

*If $T^n=I$ and $T$ is compact, then $T^n$ would be compact and so would be the identity, which is not possible in an infinite dimensional Hilbert space.


*Let $H$ be the space of square summable sequences and $T\colon (x_k)_{k\geq 1}\mapsto (\lambda_k x_{k+1})_{k\geqslant 1}$, where $\lambda_k=1$ if $k$ is even and $\lambda_k=0$ if $k$ is odd.
