Coupled Susy, pseudo-bosons and a deformed su(1, 1) Lie algebra

In a recent paper a pair of operators a and b satisfying the equations a†a = bb† + γ 1 and aa† = b†b + δ 1, has been considered, and their nature of ladder operators has been deduced and analyzed. Here, motivated by the spreading interest in non self-adjoint operators in quantum mechanics, we extend this situation to a set of four operators, c, d, r and s, satisfying dc = rs + γ 1 and cd = sr + δ 1, and we show that they are also ladder operators. We show their connection with biorthogonal families of vectors and with the so-called D-pseudo bosons. Some examples are discussed.