LAUSR

Diagana (Handbook on operator theory. Springer, Basel, pp 875–880, 2015) studied some sufficient conditions such that if S,  T and K are three unbounded linear operators with S being a closed operator, then their algebraic sum $$S+T+K$$S+T+K is also a closed operator. The main focus of this paper is to extend these results to the closable operator by adding a new concept of the gap and the $$\gamma $$γ-relative boundedness inspired by the work of Jeribi et al. (Linear Multilinear Algebra 64:1654–1668, 2015). After that, we apply the obtained results to study the specific properties of some block operator matrices.