Characterizations Of Centralizers And Lie Centralizers On B(X)

In this paper, we mainly give a local characterization of centralizer on B(H) and characterize Lie centralizer by acting zero point, idempotent operator and Jordan zero point on B(X). The details are as follows:In Chapter 1, we give some common symbols, definition of centralizer, and Lie centralizer and so on.In Chapter 2, we mainly study centralizer on B(H) by using the local properties of linear map on certain subset. we prove that if the linear mappings φ satisfying 2φ(P)=Pφ(P)+φ(P)P for all the idempotent operator of B(H), then it is a centralizer.In Chapter 3, we mainly characterize Lie centralizer on B(X) by using the local properties of linear map can be completely determined by the acting of zero product, idempotent operator product and Jordan zero product. we prove that if linear mappings φ satisfying the equation of Lie centralizer at zero point, non-trivial idempotent operator and Jordan zero point on B(X), where the dimension of Banach space X is greater than 2, then φ(A)=λA+h(A) for all A€B(X), where λ∈F and h:B(X) â†'FI is an linear map vanishing at commutators [A,B] with AB=0(resp.AB=P,Aâ—‹B=0).