Abstract

MUENTES ACEVEDO, JEOVANNY DE JESUS. On the continuity of the map square root of nonnegative isomorphisms in Hilbert spaces. Integración - UIS [online]. 2015, vol.33, n.1, pp.11-26. ISSN 0120-419X.

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 be the set of nonnegative isomorphisms in L(H). First we will show that is a convex (real) Banach manifold. Denoting by L^½ the nonnegative square root of L. In [3], Richard Bouldin proves that L^½ depends continuously on L (this proof is nontrivial). 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 selfadjoint 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^½ depends continuously on L, and moreover, he shows that the map is a homeomorphism

Keywords : Nonnegative operators; functions of operators; Hilbert spaces; spectral theory.

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