Let $n\in\mathbb N$. It is customary to call an $n\times n$ matrix $A$ self-adjoint iff the (complex conjugate of the) transpose of $A$ is equal to $A$, and to call $A$ skew-adjoint iff the (complex conjugate of the) transpose of $A$ is equal to $-A$.
Now let's go to the infinite-dimensional case: Let $H$ be a Hilbert space over $\mathbb C$ with the inner product $\langle\cdot,\cdot\rangle$ and let $T:D(T)\to H$ be a linear operator, where $D(T)\subset H$ is dense.
It is customary to call $T$ self-adjoint iff
- $\langle Tx, y\rangle = \langle x, Ty\rangle$ for all $x,y\in D(T)$; and
- $\{y\in H:\text{ The operator }T_y:D(T)\to\mathbb C, x\mapsto \langle Tx, y\rangle\text{ is bounded}\} = D(T)$.
My question. Is there an analogous notion for skew-adjoint?
My idea. An operator $T$ is called skew-adjoint iff $i T$ is self-adjoint.