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

zbMATH Open1328.47016arXiv1504.02785MaRDI QIDQ3451973

Jeovanny de Jesus Muentes Acevedo

Publication date: 18 November 2015

Abstract: Let H be a real (or complex) Hilbert space. Every nonnegative operator $LinL(H)$ admits a unique nonnegative square root $RinL(H)$, i.e., a nonnegative operator $RinL(H)$ such that $R^2=L$. Let $G{L}_S^{+}(H)$ be the set of nonnegative isomorphisms in $L(H)$. First we will show that $G{L}_S^{+}(H)$ is a convex (real) Banach manifold. Denoting by $L^{1/2}$ the nonnegative square root of $L$. In [10], Richard Bouldin proves that $L^{1/2}$ 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 $L^{1/2}$ depends continuously on $L$, and moreover, he shows that the map �egin{align}R &: GL^{+}_{S}(H) ightarrow GL^{+}_{S}(H)\ L & o L^{1/2} end{align} is a homeomorphism.

Full work available at URL: https://arxiv.org/abs/1504.02785

zbMATH Keywords

Hilbert spacesspectral theoryfunctions of operatorsnonnegative operators

Mathematics Subject Classification ID

Set-valued maps in general topology (54C60) Infinite-dimensional holomorphy (46G20) Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones) (47A56)

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