Description of bilinear forms generated by indefinite vector measures (Q735897)

Let \((H,[\cdot,\cdot])\) be a Kreĭin space and \(\xi\) be an indefinite \(J\)-orthogonal measure on the logic of all \(J\)-projectors in \(H\) with values in a complex Krein space \((K,[\cdot,\cdot]_K)\). Here, a \(J\)-selfadjoint projection \(P\) is a bounded linear idempotent operator (\(P^2=P\)) which is selfadjoint with respect to the indefinite inner product \([\cdot,\cdot]\), and a \(K\)-valued measure \(\xi\) is \(J\)-orthogonal if \(PQ=0\) implies \([\xi(P),\xi(Q)]_K=0\) for \(J\)-selfadjoint projection \(P\) and \(Q\) in \(H\). The main result says that \(\xi\) admits an extension \(\overline{\xi}\) to a bounded linear operator defined on the space of all bounded operators in \(H\) and that the indefinite measure \(\xi\) generates a \(J\)-correlation function on the space of all bounded operators in \(H\) by the formula \[ [\overline{\xi}(A), \overline{\xi}(B)]_K=\text{tr}\, ( B^+AM' + AB^+M''), \] where \(M'\) and \(M''\) are \(J\)-positive nuclear bounded operators in \(H\) and \(B^+\) denotes the adjoint of \(B\) with respect to the indefinite inner product \([\cdot,\cdot]\) in \(H\).