the set of compact operators on $H$ is nonunital

Question

How to prove $K(H)$ is a nonunital ,where $K(H)$ is the set of compact operators on $H$,$H$ is a infinite dimensional Hilbert space? Can anyone give me some hints?Thanks

Answer 1

Let $(x_n)_n$ be an orthonormal sequence in $H$. Prove that $\text{Id}(x_n) = x_n$ has no convergent subsequence.

Answer 2

For a projection $P\in B(H)$ to be compact, it has to be finite-rank. This is because its rank is $PH$, and the unit ball of a subspace will be compact if and only if said subspace is finite-dimensional.

Since $I$ is also a projection, it can only be compact when its image is finite-dimensional; that is, when $H$ is finite-dimensional.