Products of positive operators

TL;DR

This paper explores the structure and spectral properties of operators on infinite-dimensional Hilbert spaces that can be expressed as the product of two positive operators, revealing complex relationships beyond finite-dimensional cases.

Contribution

It characterizes the class of operators as products of positive operators in infinite dimensions and investigates their spectral and structural properties.

Findings

Operators in ${ mf L}^{+2}$ are connected to quasi-similarity and quasi-affinity to positive operators.

Spectral properties of these operators are developed and analyzed.

Membership criteria for algebraic and compact operators in ${ mf L}^{+2}$ are established.

Abstract

On finite dimensional spaces, it is apparent that an operator is the product of two positive operators if and only if it is similar to a positive operator. Here, the class ${\mathcal L}^{+2}$ of bounded operators on separable infinite dimensional Hilbert spaces which can be written as the product of two bounded positive operators is studied. The structure is much richer, and connects (but is not equivalent to) quasi-similarity and quasi-affinity to a positive operator. The spectral properties of operators in ${\mathcal L}^{+2}$ are developed, and membership in ${\mathcal L}^{+2}$ among special classes, including algebraic and compact operators, is examined.