In this paper, we systematically develop the “ghost” symmetry of the Kadomtsev-Petviashvili sub-hierarchy of B type (BKP) through its actions on the Lax operator L, the eigenfunctions, and the τ function. In this process, the spectral representation of the eigenfunctions and a new potential are introduced by using squared eigenfunction potential of the BKP hierarchy. Moreover, the bilinear identity of the constrained BKP hierarchy and Adler–Shiota–van Moerbeke formula of the BKP hierarchy are rederived compactly by means of the spectral representation and “ghost” symmetry.

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.