Similarity of a triangular operator to a diagonal one

A series of sufficient conditions are given for the similarity of the nonselfadjoint operator $A=G+iV^{1/2}JV^{1/2}$ (with a well-defined imaginary part) to a selfadjoint operator. Next, sufficient conditions (becoming also necessary in the dissipative case) are given for the triangular operator $f\mapsto\alpha(x)f(x)+ i\int^1_x k(x,t)f(t)d\mu(t)$ to be similar to a selfadjoint operator.