Positive elementary operators compressing spectrum

摘要

LET H be an infinite-dimensional complex Hilbert space with inner product < centre dot , centre dot > and B(H) the von Neumann algebra of all bounded linear operators on H. For T∈ B(H), σ(T), as usual, will denote the spectrum of T. Let Φ be a linear map from B(H) into itself. Φ is spectrum-preserving if σ(Φ(T) ) = σ ( T) for all T∈ B(H); Φ is spectrum-compressing if σ(Φ(T)) in contained in σ(T) for all T∈ B(H). It is clear that if Φ is unital (i.e. Φ(I) = I), then Φ is spectrum-preserving (spectrum-compressing) if and only if Φ preserves invertibility in both directions (preserves invertibility), i.e. Φ(T) is invertible if and only if T is (Φ(T) is in-vertible if T is) . Spectrum-preserving linear maps have been studied by some authors. In fact, this is one of the so-called linear preserver problems.