Maps On Kernel Infinite Idempotent Operators On Banach Spaces

Let X be an infinite-dimensional Banach space, τ∞(X)={P∈B(X): P2=P, dim KerP=∞}. For P, Q∈τ∞(X), define the order relation: P≤Q if and only ifP Q=QP; the orthogonal relation: P⊥Q if and only if P Q=QP=0. In this paper,we study bijectve maps between τ∞(X) which preserve in both dirctions the order orthe orthogonality or the commutativity, and prove that those maps almost are similartransformations.