Cyclic kernels of elementary operators on Banach spaces

Abstract

Let Λ:B(X)→B(X), Λ(S)=∑j=0n-1AjSBJ 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 Λ∗(S)=∑j=0n-1Aj∗SBj∗ be its generalized adjoint operator, where Aj∗=Hj-iKj. Let kerCΛ={S∈B(X)|∑j=0n-1AjSBj+k=0,forallk}, 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∗.