Could the kernel be of odd dimension

Question

Let $\mathscr{H}$ be a complex separable Hilbert space and $A:\mathscr{H}\oplus \mathscr{H}^* \to \mathscr{H}\oplus \mathscr{H}^*$ be a self-adjoint operator which is diagonalizable by an orthonormal basis. Let $J: \mathscr{H} \to \mathscr{H}^*$ be the anti-unitary operator defined by $(Jx)(y) = \langle x|y\rangle$ and $U:\mathscr{H}\oplus \mathscr{H}^* \to \mathscr{H}\oplus \mathscr{H}^*$ be the anti-unitary operator defined by $$ U = \begin{bmatrix} 0 &J^* \\ J &0 \end{bmatrix} $$ If $UAU^* = -A$, then is it possible that $\ker A$ is of odd dimension?