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

Abstract

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

R : GL+S (H) → GL+S (H)
L → L1/2

is a homeomorphism.

To cite this article: J.J. Muentes Acevedo, On the continuity of the map square root of nonnegative isomorphisms in Hilbert spaces, Rev. Integr. Temas Mat. 33 (2015), no. 1, 11-26.