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, H jKj=KjHj, let Λ(S) =∑j=0n-1AjSBjbe 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. © 2012 Elsevier Inc. All rights reserved.