The numerical range and the spectrum of a product of two orthogonal projections
type: Item
instance of: scholarly article
title: The numerical range and the spectrum of a product of two orthogonal projections
zbMATH Open document ID: 1321.47010
author: Hubert Klaja
published in: Journal of Mathematical Analysis and Applications
publication date: 2015-03-27T00:00:00Z
full work available at URL: https://arxiv.org/abs/1212.1996
review text: Let \(H\) be a separable complex Hilbert space. For a closed subspace \(M\) of \(H\), let \(P_M\) denote the orthogonal projection onto \(M\). The author describes \(\overline{W(P_{M_2}P_{M_1})}\), the closure of the numerical range of the product of two orthogonal projections, as a closed convex hull of the union of elliptical discs with foci at \(0\) and \(\lambda \in \sigma(P_{M_2}P_{M_1})\), and minor axis length \(\sqrt{\lambda(1-\lambda)}\). Under the additional assumption that the positive operator \(P_{M_1}P_{M_2}P_{M_1}\) is diagonalizable, the numerical range of \(P_{M_2}P_{M_1}\) (without the closure) is given as the convex hull of the union of the same ellipses as before, but this time parameterized by the point spectrum of \(P_{M_2}P_{M_1}\). The relationship between the numerical radius of a product of two orthogonal projections and its spectral radius is studied as well. The author also proves that \(W(P_{M_2}P_{M_1})\) is included in a sector with vertex at \(1\) and whose angular opening can be expressed by the cosine of the Friedrichs angle between the subspaces \(M_1\) and \(M_2\). Some other interesting results are presented. For instance, an inverse problem is considered: the information about the spectrum of \(P_{M_2}P_{M_1}\) is obtained from its numerical range. By counterexamples, two open questions posed by \textit{M. Nees} [Integral Equations Oper. Theory 35, No. 1, 85--92 (1999; Zbl 0935.47002)] are answered negatively. Finally, the author presents two applications of the obtained results. One is related to the rate of convergence in the method of alternating projections and the other to the uncertainty principle in harmonic analysis.
reviewed by: Cristina Diogo
Mathematics Subject Classification ID: 47A12
zbMATH DE Number: 6420011
zbMATH Keywords: numerical range
zbMATH Keywords: orthogonal projections
zbMATH Keywords: Friedrichs angle
zbMATH Keywords: method of alternating projections
zbMATH Keywords: uncertainty principle
zbMATH Keywords: annihilating pair
MaRDI profile type: Publication
OpenAlex ID: W2951631660
arXiv ID: 1212.1996
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DOI: 10.1016/J.JMAA.2013.09.024
description: scientific article