An operator T on a complex Hilbert spaces \(\mathfrak{H}\) is said to be skew symmetric if there exists a conjugate-linear isometric \(C:\mathfrak{H}\to \mathfrak{H}\) so that \(C T C=-T^*\). In view of various results on complex symmetric operators, the authors give a characterization of normal operators which are skew symmetric. Their main results are as follows.
[*] If \(T\in B(\mathfrak{H})\) is normal, then the following are equivalent:
[i] \(T\) is skew symmetric.
[ii] \(T|_{(\mathrm{ker} T)^\perp}\simeq N\oplus(-N)\), where \(N\) is a normal operator on some Hilbert space \(\mathfrak{K}\) with \(E_N(\mathbb{C}- \Sigma)=0\).
[iii] \(T|_{(\mathrm{ker} T)^\perp}\simeq N\oplus(-N)\), where \(N\) is a normal operator on some Hilbert space \(\mathfrak{K}\).
[**] A normal operator \(T\in B(\mathfrak{H})\) is skew symmetric if and only if there are mutually singular measures \(\mu_{\infty}, \mu_{1}{}, \mu_{2}{},\dots\) (some of which may be zero) such that:
[i] \(\mu_j(\delta)=\mu_j(-\delta)\) for any \(1\leq j\leq\infty\) and any Borel subset \(\delta\) of \(\mathbb{C}\).
[ii] \(T\) is unitarily equivalent to the operator \[ N=\bigoplus_{1\leq j\leq\infty}N_j^{(j)}, \] where \(N_j\) is the “multiplication by z” operator on \(L^2(\mu_j)\; (1\leq j\leq \infty)\).