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.
Sobre la continuidad de la aplicación raíz cuadrada de
isomorfismos no negativos en espacios de Hilbert
Resumen. Sea H un espacio de Hilbert real (o complejo). Todo operador no
negativo L ∈ L(H) admite una única raíz cuadrada no negativa R ∈ L(H),
esto es, un operador no negativo R ∈ L(H) tal que R2 = L. Sea
el conjunto de los isomorfismos no negativos en L(H). Primero probaremos
que
es una variedad de Banach (real). Denotando como L½ la raíz
cuadrada no negativa de L, en [3] Richard Bouldin prueba que L½ depende
continuamente de L (esta prueba es no trivial). Este resultado tiene varias
aplicaciones. Por ejemplo, es usado para encontrar la descomposición polar de
un operador limitado. Esta descomposición polar nos lleva a determinar los
subespacios espectrales positivos y negativos de cualquier operador autoadjunto,
y además, lleva a definir el índice de Máslov. El autor de este artículo
da una prueba alternativa (y un poco más simplificada) de que L½ depende
continuamente de L, y además, prueba que la aplicación
es un homeomorfismo.
Palabras claves: Operadores no negativos, funciones de operadores, espacios de Hilbert, teoría espectral.
Texto Completo disponible en PDF
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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.