CHARACTERIZATIONS OF JORDAN †-SKEW MULTIPLICATIVE MAPS ON OPERATOR ALGEBRAS OF INDEFINITE INNER PRODUCT SPACES

Abstract

Let H and K be indefinite inner product spaces. This paper shows that a bijective map $\Phi: {\mathcal B}(H) \rightarrow {\mathcal B}(K)$ satisfies Φ(AB† + B†A) = Φ(A)Φ(B)† + Φ(B)†Φ(A) for every pair $A,B \in {\mathcal B}(H)$ if and only if either Φ(A) = cU AU† for all A or Φ(A) = cU A†U† for all A; Φ satisfies Φ(AB† A) = Φ(A)Φ(B)†Φ(A) for every pair $A,B \in {\mathcal B}(H)$ if and only if either Φ(A) = U AV for all A or Φ(A) = U A†V for all A, where A† denotes the indefinite conjugate of A, U and V are bounded invertible linear or conjugate linear operators with U†U = c-1I and V†V = cI for some nonzero real number c.