Let X be a complex Banach space. If 8: B(X) Ä B(X) is a surjective linear map such that A and 8(A) have the same spectral radius for every A # B(X), then 8=c3 where 3 is either an algebra-automorphism or an antiautomorphism of B(X) and c is a complex constant such that |c|=1.
Linear Maps Preserving Operators of Local Spectral Radius Zero
✍ Scribed by Constantin Costara
- Book ID
- 113014246
- Publisher
- SP Birkhäuser Verlag Basel
- Year
- 2012
- Tongue
- English
- Weight
- 224 KB
- Volume
- 73
- Category
- Article
- ISSN
- 0378-620X
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📜 SIMILAR VOLUMES
Let L(H) be the algebra of all bounded linear operators on an infinite dimensional complex Hilbert H. We characterize linear maps from L(H) onto itself that preserve the essential spectral radius.
Define the sign-real spectral radius of a real n × n matrix A as ρ s 0 (A) = max S∈S ρ 0 (SA), where ρ 0 (A) = max{|λ|; λ a real eigenvalue of A} is the real spectral radius of A and S denotes the set of signature matrices, i.e. S = {S; |S| = I}, the absolute value of matrices being meant entrywise.
Let H be a complex Hilbert space of dimension greater than 2 and J ∈ B(H) be an invertible self-adjoint operator. Denote by A † = J -1 A \* J the indefinite conjugate of A ∈ B(H) with respect to J and denote by w(A) the numerical radius of A. Let W and V be subsets of B(H) which contain all rank one