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 such that Φ(T)=cUTU[∗] for all T∈L(H1) with UU[∗]=cI2, U[∗]U=cI1 and c=±1. Here I1 and I2 denote the identity maps on H1 and H2, respectively.