Rank-preserving linear maps on B(X) (Q5899722)

Let X be a Banach space with dim X\(>2\), and B(X) the Banach algebra of all bounded linear operators on X. Let \(\phi\) be a linear map from B(X) to B(X). \(\phi\) is called to be rank-preserving if it preserves the rank of every finite rank operator. The author proved that Theorem. If \(\phi\) is a weakly continuous linear map on B(X), then \(\phi\) is rank-preserving if and only if either i) there is an injective operator A in B(X) and an operator B in B(X) with dense range such that \(\phi (T)=ATB\) for every T in B(X); or ii) there is an injective operator A in \(B(X^*\to X)\) and an operator B in \(B(X\to X^*)\) with dense range such that \(\phi (T)=AT^*B\) for every T in B(X).