Hermitian indices and factorization of selfadjoint operators on a Kreĭn space

Abstract

The hermitian indices of a selfadjoint operator $C$ on a Kreĭn space $\mathcal{H}$ are defined as geometric measures of positivity and negativity of the operator. A different pair of indices arises in the Bognár-Krámli factorization of $C$, which writes $C$ as a product $A{A}^{\ast }$ where $A$ acts on a Kreĭn space $\mathcal{A}$ into $\mathcal{H}$ and has zero kernel; the new indices are the positive and negative indices of $\mathcal{A}$. Such factorizations are far from unique. When $\mathcal{H}$ is separable, it is known that the two notions of indices always coincide, and this has applications to index formulas in the theory of Julia operators and completion problems for operator matrices. A new proof of the equality of indices that does not require separability is given in this work.