Spectral enclosures for a class of block operator matrices

We prove new spectral enclosures for the non-real spectrum of a class of $2\times 2$ block operator matrices with self-adjoint operators $A$ and $D$ on the diagonal and operators $B$ and $-B^{\ast }$ as off-diagonal entries. One of our main results resembles Gershgorin’s circle theorem. The enclosures are applied to $J$-frame operators.