Bijectivity of -endomorphisms of ℬ( H) and the unilateral shift.

Let H be a separable infinite-dimensional complex Hilbert space. We prove that if Ф: ℬ( H)→ℬ( H) is a-preserving ring homomorphism whose range contains a rank-one operator and an operator with dense range, then Ф is an isometric linear or conjugate-linear algebra automorphism of ℬ( H). In particular, if the unilateral shift is contained in the range of a-endomorphism Ф of ℬ( H), then Ф is bijective. [ABSTRACT FROM AUTHOR]