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The effect of perturbations of linear operators on their polar decomposition

Authors: Richard Duong and Friedrich Philipp
Journal: Proc. Amer. Math. Soc.
MSC (2010): Primary 47A05; Secondary 47A55
Published electronically: August 22, 2016
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Abstract: The effect of matrix perturbations on the polar decomposition has been studied by several authors and various results are known. However, for operators between infinite-dimensional spaces the problem has not been considered so far. Here, we prove in particular that the partial isometry in the polar decomposition of an operator is stable under perturbations, given that kernel and range of original and perturbed operator satisfy a certain condition. In the matrix case, this condition is weaker than the usually imposed equal-rank condition. It includes the case of semi-Fredholm operators with agreeing nullities and deficiencies, respectively. In addition, we prove a similar perturbation result where the ranges or the kernels of the two operators are assumed to be sufficiently close to each other in the gap metric.

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Richard Duong
Affiliation: Institut für Mathematik, Technische Universität Berlin, Straße des 17. Juni 136, 10623 Berlin, Germany

Friedrich Philipp
Affiliation: Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria, Pabellón I, 1428 Buenos Aires, Argentina

Keywords: Linear operator, Hilbert space, polar decomposition, perturbation
Received by editor(s): March 2, 2016
Received by editor(s) in revised form: April 19, 2016
Published electronically: August 22, 2016
Communicated by: Stephan Ramon Garcia
Article copyright: © Copyright 2016 American Mathematical Society