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

JEOVANNY DE JESUS MUENTES ACEVEDO*

Universidade de São Paulo, Instituto de Matemática e Estatística, São Paulo, Brasil.

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 be the set of nonnegative isomorphisms in L(H). First we will show that is a convex (real) Banach manifold. Denoting by L½ the nonnegative square root of L. In [3], Richard Bouldin proves that L½ 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 L½ depends continuously on L, and moreover, he shows that the map

is a homeomorphism.

Keywords: Nonnegative operators, functions of operators, Hilbert spaces, spectral theory.
MSC2010: 47A56, 46G20, 54C60.

isomorfismos no negativos en espacios de Hilbert

es un homeomorfismo.

Palabras claves: Operadores no negativos, funciones de operadores, espacios de Hilbert, teoría espectral.

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*Email:jeovanny@ime.usp.br
Received: 18 September 2014, Accepted: 11 December 2014.
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.