On the continuity of the map square root of nonnegative isomorphisms in Hilbert spaces - Jeovanny de Jesus Muentes Acevedo (Revista Integración, 2015)
Let H be a real (or complex) Hilbert space. Every nonnegative operator L ∈ L(H) admits a unique nonnegative square root R ∈ L(H), i.e., a nonnegative operator R ∈ L(H) such that R2 = 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 L1/2 the nonnegative square root of L. In [3], Richard Bouldin proves that L1/2 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 L1/2 depends continuously on L, and moreover, he shows that the map is a homeomorphism.