Product of nonnegative selfadjoint operators in unbounded settings

TL;DR

This paper establishes conditions for factorizing unbounded operators into nonnegative selfadjoint parts, extending known results and introducing new inequalities and characterizations related to similarity and quasi-affinity.

Contribution

It provides new necessary and sufficient conditions for such factorizations, including cases where operators are bounded or invertible, and introduces a reversed Sebestyén inequality.

Findings

Characterization of operator classes via quasi-affinity and Sebestyén inequality

Extension of factorization results to unbounded operators

Introduction of a reversed Sebestyén inequality for invertible cases

Abstract

In this paper, necessary and sufficient conditions are established for the factorization of a closed, in general, unbounded operator $T=AB$ into a product of two nonnegative selfadjoint operators $A$ and $B.$ Already the special case, where $A$ or $B$ is bounded, leads to new results and is of wider interest, since the problem is connected to the notion of similarity of the operator $T$ to a selfadjoint one, but, in fact, goes beyond this case. It is proved that this subclass of operators can be characterized not only by means of quasi-affinity of $T^*$ to an operator $S=S^* \geq 0$, but also via Sebesty\'en inequality, a result known in the setting of bounded operators $T.$ Another subclass of operators $T,$ where $A$ or $B$ has a bounded inverse, leads to a similar analysis. This gives rise to a reversed version of Sebesty\'en inequality which is introduced in the present paper. It is shown that this second subclass, where $A^{-1}$ or $B^{-1}$ is bounded, can be characterized in a similar way by means of quasi-affinity of $T,$ rather that $T^*,$ to an operator $S=S^*\geq 0$. Furthermore, the connection between these two classes and weak-similarity as well as quasi-similarity to some $S=S^*\geq 0$ is investigated. Finally, the special case where $ S$ is bounded is considered.