A characterization for *-isomorphisms in an indefinite inner product space

Let H1 and H2 be indefinite inner product spaces. Let L (H1) and L (H2) be the sets of all linear operators on H1 and H2, respectively. The following result is proved: If Φ is [*]-isomorphism from L (H1) onto L (H2) then there exists U : H1 → H2 such that Φ (T) = c U T U[*] for all T ∈ L (H1) with U U[*] = c I2, U[*] U = c I1 and c = ± 1. Here I1 and I2 denote the identity maps on H1 and H2, respectively. © 2006 Elsevier Inc. All rights reserved.