  We study basic spectral properties of J-self-adjoint $2\times 2$ block
operator matrices. Using the linear resolvent growth condition, we obtain
simple necessary conditions for the regularity of the critical point $\infty$.
In particular, we present simple examples of operators having the singular
critical point $\infty$. Also, we apply our results to the linearized operator
arising in the study of soliton type solutions to the nonlinear relativistic
Ginzburg-Landau equation.
