Nonlinear maps preserving numerical radius of indefinite skew products of operators

• Published in: Linear algebra and its applications 2009-04, Vol.430 (8), p.2240-2253
• Main Authors: Hou, Jinchuan, He, Kan, Zhang, Xiuling
• Format: Article
• Language: English
• Subjects: Algebra; Exact sciences and technology; Indefinite skew product of operators; Linear and multilinear algebra, matrix theory; Mathematical analysis; Mathematics; Nonlinear map; Numerical radius; Operator theory; Sciences and techniques of general use
• ISSN: 0024-3795; 1873-1856
• DOI: 10.1016/j.laa.2008.12.002

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 Φ : W → V be a surjective map. We show that Φ satisfies w ( AB † ) = w ( Φ ( A ) Φ ( B ) † ) and w ( A † B ) = w ( Φ ( A ) † Φ ( B ) ) for all A , B ∈ W if and only if there exist scalars ϵ i ∈ { - 1 , 1 } ( i = 1 , 2 ) , unitary (or conjugate unitary) operators U , V on H satisfying U † U = ϵ 1 I , V † V = ϵ 2 I and a functional φ : W → C with | φ ( A ) | ≡ 1 such that Φ ( A ) = φ ( A ) UAV for all A ∈ W ; Φ satisfies w ( AB † A ) = w ( Φ ( A ) Φ ( B ) † Φ ( A ) ) for all A , B ∈ W if and only if either there exist ϵ ∈ { - 1 , 1 } , a unitary (or conjugate unitary) operator U on H satisfying U † U = ϵ I and a functional φ : W → C with | φ ( A ) | ≡ 1 such that Φ ( 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 Φ ( A ) = φ ( A ) UA ∗ U ∗ for all A ∈ W .