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 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.
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 L^½ la raíz cuadrada no negativa de L, en [3] Richard Bouldin prueba que L^½ 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 L^½ 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

References

[1] Bachman G. and Narici L., Functional Analysis, Reprint of the 1966 original. Dover Publications, Inc., Mineola, New York, 2000.

[2] Bernardes N.C. e Fernandez C.S., Introdução às funções de uma variável complexa, textos Universitários, Sociedade Brasileira de Matemática, 2008.

[3] Bouldin R., "The Norm Continuity Properties of Square Roots", SIAM J. Math. Anal. 3 (1972), 206-210.

[4] Conway J.B., Functions of One Complex Variable, Graduate Texts in Mathematics, 11, Springer-Verlag, New York-Heidelberg, 1973.

[5] Fabian M., Habala P., Hájek P., Montesinos V., Pelant J. and Zizler V., Functional Analysis and Infinite-Dimensional Geometry, CMS Books in Mathematics, 8, Springer-Verlag, New York, 2001.

6] Fitzpatrick P.M., Pejsachowicz J. and Recht L., "Spectral flow and bifurcation of critical points of strongly-indefinite functionals. I. General theory", J. Funct. Anal. 162 (1999), no. 1, 52-95.

[7] Kato T., Perturbation Theory for Linear Operators, Grundlehren der Mathematischen Wissenschaften, 132, Springer-Verlag, Berlín-New York, 1976.

[8] Knopp K., Theory of Functions, I, Elements of the General Theory of Analytic Functions, Dover Publications, New York, 1945.

[9] Kreyszig E., Introductory Functional Analysis with Applications, John Wiley & Sons, New York-London-Sydney, 1978.

[10] Long Y., "A Maslov type index theory for sympletics paths", Topol. Methods Nonlinear Anal. 10 (1997), no. 1, 47-78.

[11] Schechter M., Principles of Functional Analysis, Second Edition, Graduate Studies in Mathematics, 36, American Mathematical Society, Providence, RI, 2002.

[12] Taylor A.E., Introduction to Functional Analysis, John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London, 1958.

^*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.