  A $J$-frame is a frame $\mathcal{F}$ for a Krein space $(\mathcal{H}, [\, ,
\,])$ which is compatible with the indefinite inner product $[\, , \, ]$ in the
sense that it induces an indefinite reconstruction formula that resembles those
produced by orthonormal bases in $\mathcal{H}$. With every $J$-frame the
so-called $J$-frame operator is associated, which is a self-adjoint operator in
the Krein space $\mathcal{H}$. The $J$-frame operator plays an essential role
in the indefinite reconstruction formula.
  In this paper we characterize the class of $J$-frame operators in a Krein
space by a $2\times 2$ block operator representation. The $J$-frame bounds of
$\mathcal{F}$ are then recovered as the suprema and infima of the numerical
ranges of some uniformly positive operators which are build from the entries of
the $2\times 2$ block representation. Moreover, this $2\times 2$ block
representation is utilized to obtain enclosures for the spectrum of $J$-frame
operators, which finally leads to the construction of a square root. This
square root allows a complete description of all $J$-frames associated with a
given $J$-frame operator.
