Completions of upper-triangular matrices to Kato nonsingularity

In this paper we consider the problem of completion of the upper-triangular operator matrix [A?0B] where A∈B(H) and B∈B(K), to Kato nonsingular operators and completely solve it in each of the following cases: one of the operators A or B is Kato nonsingular; B is injective; A is with dense range; B is with finite ascent; A is with finite descent; 0∉int(σp(B)); 0∉int(σcp(A)). In particular, the results generalize and complete those of [3] and [7] that concern the same problem.