Similarity-preserving Linear Maps On B(x)

Let X be a complex Banach space with the topological dual X* .By B(X) we denotethe set of all bounded linear operators on X. We say that two operators A and B are similar,denoted by A,B, if there exists an invertible operator S' such that A = SBS-1.A mapÑ„:B(x)-B (X ) is called to be similarity-preserving if A~B implies thatÑ„(A)~Ñ„(B).A functional h in X * is called to be similarity-invariant if A~B implies that h(A) = h(B).The main result in this thesis rea,de a,e follows.Let X be an infinite-dimensional Banach space andÑ„:B(X)~ B(X) be a bijectivemap. ThenÑ„is similarity-preserving if and only if one of the following holds:(i) There exist a nonzero complex number C,an invertible bounded operator T in B(X)and a similarity-invariant linear functional h on B(X) with h(I)≠-c。such thatÑ„(A)= cTAT-1+h(A)I for all A B(x):(ii) There exist a nonzero complex number c,an invertible bounded operator T:X*â†'Xand a similarity-invariant linear functional h on B(X) with h(I)â†'-c。such thatÑ„(A)=cTA*T-1+h(A)I for all A B(X).Also, we investigate some properties of similarity-invariant functionals.