On to the continuity of the map square root of nonnegative isomorphisms in Hilbert spaces

TL;DR

This paper proves that the map taking a nonnegative isomorphism in a Hilbert space to its square root is a homeomorphism, establishing its continuity and topological properties, which simplifies previous proofs and has important applications.

Contribution

The paper provides a simplified proof that the square root map on nonnegative isomorphisms in Hilbert spaces is a homeomorphism, enhancing understanding of its topological structure.

Findings

The set of nonnegative isomorphisms forms a convex Banach manifold.

The square root map is a homeomorphism on this set.

Continuity of the square root map enables applications like polar decomposition.

Abstract

Let H be a real (or complex) Hilbert space. Every nonnegative operator $L \in L(H)$ admits a unique nonnegative square root $R \in L(H)$, i.e., a nonnegative operator $R \in L(H)$ such that $R^{2}= L$. Let $GL^{+}_{S}(H)$ be the set of nonnegative isomorphisms in $L(H)$. First we will show that $GL^{+}_{S}(H)$ is a convex (real) Banach manifold. Denoting by $L^{1/2}$ the nonnegative square root of $L$. In [10], Richard Bouldin proves that $L^{1/2}$ depends continuously on $L$ (this proof is non-trivial). This result has several applications. For example, it is used to find the polar decomposition of a bounded operator. This polar decomposition allows us to determine the positive and negative spectral subespaces of any self-adjoint operator, and moreover, allows us to define the Maslov index. The autor of the paper under review provides an alternative proof (and a little more simplified) that $L^{1/2}$ depends continuously on $L$, and moreover, he shows that the map \begin{align}R &: GL^{+}_{S}(H)\rightarrow GL^{+}_{S}(H)\\ L &\to L^{1/2} \end{align} is a homeomorphism.