The numerical range and the spectrum of a product of two orthogonal projections
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.
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