If an operator $X$ on a Hilbert space satisfies $X^*=-X$, then is $X$ equal to $0$?

Question

If a linear operator $X$ on a Hilbert space satisfies $X^*=-X$, then is $X$ equal to $0$? $X^*$ is the adjoint.

Answer 1

No. Such operators called skew-symmetric. For example the matrix

$X=\begin{bmatrix}0&2&-1\\-2&0&-4\\1&4&0\end{bmatrix}$

gives a skew-symmetric operator on $\mathbb R^3$ (or $\mathbb C^3$)