  We consider Toeplitz operators defined on a concave corner-shaped subset of
the square lattice. We obtain a necessary and sufficient condition for these
operators to be Fredholm. We further construct a Fredholm concave corner
Toeplitz operator of index one. By using this, a relation between Fredholm
indices of quarter-plane and concave corner Toeplitz operators is clarified. As
an application, topological invariants and corner states for some bulk-edges
gapped Hamiltonians on two-dimensional (2-D) class AIII and 3-D class A systems
with concave corners are studied. Explicit examples clarify that these
topological invariants depend on the shape of the system. We discuss the
Benalcazar--Bernevig--Hughes' 2-D Hamiltonian and see that there still exists
topologically protected corner states even if we break some symmetries as long
as the chiral symmetry is preserved.
