A note on the operator equation XA-AX=X~p
Hong-Ke Du · Journal of Baoji University of Arts and Sciences · 2008
Aim The operator equation XA-AX=Xp(1≤p∞,Xp≠0)was studied.Methods The proofs in this paper applied the techniques of operator theory and the block operator matrix.Results(1)The solutions of XA-AX=Xp are quasinilpotent.As a corollary,the solution X is nilpotent if the dimension of H is finite(Dietrich Burde.On the matrix equation XA-AX=Xp.Linear Algebra and Its Applications,2005,404:147-165.).(2)If X is a nilpotent solution of operator equation XA-AX=Xp for all positive integers p≥2,then XEA(σ)=EA(σ)X.Conclusion For operator equation XA-AX=Xp(1≤p∞,Xp≠0),it is sufficient to consider the case that σ(A) is single connected,if we only study the properties of its nilpotent solution.