Let \(L(\lambda)=\lambda^ nA_ 0+...+A_ n\) be an operator pencil on a Hilbert space H with \(A_ 0,...,A_ n\) selfadjoint and \(A_ 0\) invertible. Then L(\(\lambda)\) can be associated with a pencil \(\tilde L(\lambda)= \lambda\tilde I-\tilde L\) with the same spectrum, defined on a direct sum \(\tilde H\) of n copies of H. The author shows that if \(L(\lambda)\) is completely hyperbolic (i.e. it satisfies a certain condition in terms of the Bezout matrix), then there exists a strictly positive operator S (which can be explicitly constructed) such that \((\tilde S\tilde L)^*=\tilde S\tilde L\).