If the only self adjoint operators from $\mathcal H$ to $\mathcal H$ are $0$, what comment we can make on $\mathcal H$?

Question

If $\mathcal H$ is a Hilbert space over $\mathbb C $ or $\mathbb R$.

The only self adjoint operators from $\mathcal H$ to $\mathcal H$ are $0$ and $I$. --------------------------------(1)

1. Then can we say that $\mathcal H$ is a one-dimensional space?

2. What other comments we can make on (1).

Answer 1

The operator $2 \, I$ is always self-adjoint. Hence, $2 \, I = I$ or $2 \, I = 0$. This yields $I = 0$, hence, $\mathcal{H} = \{0\}$.

Edit: Since every orthogonal projection onto a subspace is self-adjoint, it is quite easy to reconstruct $\mathcal{H}$ from its self-adjoint operators.