  We estimate the size of the spectral gap at zero for some Hermitian block
matrices. Included are quasi-definite matrices, quasi-semidefinite matrices
(the closure of the set of the quasi-definite matrices) and some related block
matrices which need not belong to either of these classes. Matrices of such
structure arise in quantum models of possibly disordered systems with
supersymmetry or graphene like symmetry. Some of the results immediately extend
to infinite dimension.
