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 $AA^*$ 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.

Theorems & Definitions (21)

• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Definition 2.3
• Theorem 2.4
• Corollary 2.5
• proof : Proof of Theorem $\ref{['indices-exist']}$
• Theorem 2.6: Sylvester theorem
• proof