An extension and refinement of the theorems of Douglas and Sebesty\'en for unbounded operators

TL;DR

This paper extends and refines Douglas and Sebestyén's theorems for unbounded operators, providing new necessary and sufficient conditions for operator factorizations involving bounded nonnegative operators.

Contribution

It introduces a novel extension of Sebestyén's theorem to unbounded operators and refines Douglas's factorization theorem with new conditions involving intermediate selfadjoint operators.

Findings

Established conditions for factorization with bounded nonnegative operators.

Extended Sebestyén's theorem to unbounded operators.

Provided criteria for intermediate selfadjoint operators H.

Abstract

For a closed densely defined operator $T$ from a Hilbert space $\mathfrak{H}$ to a Hilbert space $\mathfrak{K}$, necessary and sufficient conditions are established for the factorization of $T$ with a bounded nonnegative operator $X$ on $\mathfrak{K}$. This result yields a new extension and a refinement of a well-known theorem of R.G. Douglas, which shows that the operator inequality $A^*A\leq \lambda^2 B^*B, \lambda \geq 0$, is equivalent to the factorization $A=CB$ with $\|C\|\leq \lambda$. The main results give necessary and sufficient conditions for the existence of an intermediate selfadjoint operator $H\geq 0$, such that $A^*A \leq \lambda H \leq \lambda^2 B^*B$. The key results are proved by first extending a theorem of Z. Sebesty\'en to the setting of unbounded operators.