Linear maps between operator algebras preserving certain spectral functions

Abstract

Let H be an infinite dimensional complex Hilbert space and let φ be a surjective linear map on B(H) with φ(I)−I ∈ K(H), where K(H) denotes the closed ideal of all compact operators on H. If φ preserves the set of upper semi-Weyl operators and the set of all normal eigenvalues in both directions, then φ is an automorphism of the algebra B(H). Also the relation between the linear maps preserving the set of upper semi-Weyl operators and the linear maps preserving the set of left invertible operators is considered.

Additive maps preserving semi-Fredholm operators with bounded ascent on $\mathcal B(\mathcal X)$

“…This problem was solved in some special cases of semi-simple Banach algebras ( [9], [12], [16], [22]). Several authors have studied linear maps which preserve the classes of semi-Fredholm operators, Fredholm operators, and the related operators in both directions ( [1], [4], [7]). It has been shown that such maps preserve the ideal of compact operators in both directions, and the maps induced by them on the Calkin algebra are Jordan automorphisms.…”

Additive maps preserving the semi-Fredholm domain in spectrum

“…It is known that the notion of semi-Fredholm operators have close relationship with compact operators. Several authors have studied the linear maps which preserve Fredholm operators, semi-Fredholm operators, or upper semi-Weyl operators in both directions (see [2], [8]). They showed that such maps preserve the ideal of compact operators in both directions, and their induced maps on the Calkin algebra are Jordan automorphisms.…”

“…They showed that such maps preserve the ideal of compact operators in both directions, and their induced maps on the Calkin algebra are Jordan automorphisms. For example, in [2], the authors discuss the linear surjective maps preserving upper semi-Weyl operators, and show that their induced maps on the Calkin algebra are Jordan automorphisms. However, the problem of determining the structure of the maps itself hasn't been solved.…”