Cyclic kernels of elementary operators on Banach spaces

Let Λ:B(X)→B(X), be elementary operator, where B(X) is the algebra of all bounded linear operators on a Banach space. For Aj and Bj prenormal, i.e. Aj=Hj+iKj, with ‖exp(itHj)‖, ‖expitKj)‖ bounded, HjKj=KjHj, let be its generalized adjoint operator, where . Let , where for j⩾n we take Bj=Bj-n. We prove that for two commutative families of prenormal operators Aj,Bj, there holds kerCΛ=kerCΛ∗.