Compact Operator Inversion

Question

Let I be a positive compact in $\mathscr{B}(\mathscr{H})$ (where $\mathscr{H}$ is some Hilbert space) then $I$ can be written (uniquely) as $A^2=I$ for some $A \in \mathscr{B}(\mathscr{H})$.

My question is: "is $A$ necessarily invertible?"

Answer 1

If the Hilbert space is infinite dimensional, then $A$ is necessarily non-invertible, because if $A$ is invertible, so is $I$, but $I$ is compact and compact operators can not be invertible in Infinite dimensional Banach spaces.

If the Hilbert space is finite dimensional, then $I$ is injective (because it is positive), hence $A$ is also injective, hence bijective (for reasons of finite dimension), hence invertible.