Abstract
We investigate connections between the geometry of linear subspaces and the convergence of the alternating projection method for linear projections. The aim of this article is twofold: in the first part, we show that even in Euclidean spaces the convergence of the alternating method is not determined by the principal angles between the subspaces involved. In the second part, we investigate the properties of the Oppenheim angle between two linear projections. We discuss, in particular, the question of existence and uniqueness of “consistency projections” in this context.
Highlights
The interest in the convergence of sequences of iterates of projections of various types goes back at least to the mid-twentieth century
Bargetz et al / Linear Algebra and its Applications 603 (2020) 41–56 can be considered one of the starting points of these investigations. In this article he shows that given a Hilbert space H and two closed subspaces M, N ⊂ H, with corresponding orthogonal projections PM and PN, respectively, the sequence defined by x0 ∈ H
Oppenheim introduced in [19] an angle between linear projections in Banach spaces
Summary
The interest in the convergence of sequences of iterates of projections of various types goes back at least to the mid-twentieth century. Bargetz et al / Linear Algebra and its Applications 603 (2020) 41–56 can be considered one of the starting points of these investigations In this article he shows that given a Hilbert space H and two closed subspaces M, N ⊂ H, with corresponding orthogonal projections PM and PN , respectively, the sequence defined by x0 ∈ H x2n+1 = PM x2n and x2n+2 = PN x2n+1 converges in norm to PM∩N x0 for every initial point x0 ∈ H. Oppenheim introduced in [19] an angle between linear projections in Banach spaces This concept was developed further in [20], where a number of sufficient conditions for convergence of iterates of projections are given. We investigate the properties of the Oppenheim angle between two linear projections and provide an example which shows that the modification of the definition of this angle introduced in [20] is necessary
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