Nonlinear maps preserving numerical radius of indefinite skew products of operators

✍ Scribed by Jinchuan Hou; Kan He; Xiuling Zhang

• Publisher: Elsevier Science
• Year: 2009
• Tongue: English
• Weight: 215 KB
• Volume: 430
• Category: Article
• ISSN: 0024-3795
• DOI: 10.1016/j.laa.2008.12.002

✦ Synopsis

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 operators, and let : W → V be a surjective map. We show that satisfies w(AB † ) = w( (A) (B) † ) and w(A † B) = w( (A) † (B)) for all A, B ∈ W if and only if there exist scalars i ∈ {-1, 1}(i = 1, 2), unitary (or conjugate unitary) operators U, V on H satisfying U † U = 1 I, V † V = 2 I and a functional ϕ :

and only if either there exist ∈ {-1, 1}, a unitary (or conjugate unitary) operator U on H satisfying U † U = I and a functional ϕ : W → C with |ϕ(A)| ≡ 1 such that (A) = ϕ(A)UAU * for all A ∈ W; or, there exist a nonzero real number b, a unitary (or conjugate unitary) operator U on H satisfying U * JU = bJ -1 and a functional ϕ : W → C with |ϕ(A)| ≡ 1 such that (A) = ϕ(A)UA * U * for all A ∈ W.