Hermitian indices and factorization of selfadjoint operators on a Kre\u{i}n space

TL;DR

This paper explores the relationship between hermitian indices and factorization indices of selfadjoint operators on Kre space, providing a new proof of their equality without assuming separability.

Contribution

It introduces a novel proof showing the equality of hermitian and factorization indices for selfadjoint operators on Kre spaces without the separability assumption.

Findings

Hermitian and factorization indices coincide for selfadjoint operators.

A new proof of index equality that does not rely on separability.

Applications to Julia operators and operator matrix completion.

Abstract

The hermitian indices of a selfadjoint operator $C$ on a Kre\u{i}n space $\mathcal H$ are defined as geometric measures of positivity and negativity of the operator. A different pair of indices arises in the Bogn\'ar-Kr\'amli factorization of $C$, which writes $C$ as a product $AA^*$ where $A$ acts on a Kre\u{i}n 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.