The gauge transformation of the modified KP hierarchy

In this paper, we firstly investigate the successive applications of three elementary gauge transformation operators T_i with i = 1,2,3 for the mKP hierarchy in Kupershmidt-Kiso version, and find that the gauge transformation operators T_i can not commute with each other. Then two types of gauge transformation operators T_D and T_I constructed from T_i are proved that they can commute with each other. In particular, T_I is introduced for the first time in the literature. And the successive applications of T_D and T_I in the form of T^(n,k), which is the product of n terms of T_D and k terms of T_I, are derived in three cases for different n and k. At last, the corresponding successive applications of T_D and T_I on the eigenfunction Φ, the adjoint eigenfunction Ψ and the tau functions τ₀ and τ₁ are considered.