Nonlinear maps preserving numerical radius of indefinite skew products of operators

AbstractLet 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-1A∗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 ϵi∈{-1,1} (i=1,2), unitary (or conjugate unitary) operators U, V on H satisfying U†U=ϵ1I, V†V=ϵ2I 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=ϵI 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