Ritt operators and convergence in the method of alternating projections

Abstract

Abstract. Given N ≥ 2 closed subspaces M1, . . . , MN of a Hilbert space X, let P k denote the orthogonal projection onto M k , 1 ≤ k ≤ N . It is known that the sequence (xn), defined recursively by x0 = x and xn+1 = PN · · · P1xn for n ≥ 0, converges in norm to PM x as n → ∞ for all x ∈ X, where PM denotes the orthogonal projection onto M = M1 ∩ . . . ∩ MN . Moreover, the rate of convergence is either exponentially fast for all x ∈ X or as slow as one likes for appropriately chosen initial vectors x ∈ X. We give a new estimate in terms of natural geometric quantities on the rate of convergence in the case when it is known to be exponentially fast. More importantly, we then show that even when the rate of convergence is arbitrarily slow there exists, for each real number α > 0, a dense subset Xα of X such that xn − PM x = o(n −α ) as n → ∞ for all x ∈ Xα. Furthermore, there exists another dense subset X∞ of X such that, if x ∈ X∞, then xn −PM x = o(n −α ) as n → ∞ for all α > 0. These latter results are obtained as consequences of general properties of Ritt operators. As a by-product, we also strengthen the unquantified convergence result by showing that PM x is in fact the limit of a series which converges unconditionally.

The optimal error bound for the method of simultaneous projections

“…In this section we present an argument showing that Theorem 4 which is stated here for a real Hilbert space indeed follows from [3,Theorem 4.3], which is shown to be true for a complex Hilbert space. The proof is based on a hand-written note which has kindly been provided to us by Catalin Badea and David Seifert.…”

“…We remark here that Theorem 4 follows from [3,Theorem 4.3] which was only proved for a complex Hilbert space. In order to see this, one can apply a complexification argument which has kindly been provided to us by Catalin Badea and David Seifert; see the Appendix for more details.…”

“…Alternative (ii) of the above theorem has recently been extended in the case of the cyclic projection method. Following [3], we say that T k converges super-polynomially fast to T ∞ on a nonempty set X ⊆ H if T k (x) − T ∞ (x) /k −α → 0 as k → ∞ for all α > 0 and x ∈ X.…”

“…In addition, one can show that cos(M 1 , M 2 ) < 1 if and only if M ⊥ 1 + M ⊥ 2 is closed; see, for example, [11,Theorem 9.35] and [11, p. 235] for a complete proof and detailed historical notes going back to [6,10] and Simonič. As far as we are aware, for r > 2 the exact computation of the error operator norm for both algorithmic operators is still unknown; see, for example, [2,3,14,15,21,22]. Even for r = 2, the norm ((P M1 + P M2 )/2) k − P M seems to be unknown.…”

Non-optimality of the Greedy Algorithm for Subspace Orderings in the Method of Alternating Projections

“…In view of these applications it is important to understand the rate at which the convergence in (1.1) takes place; see for instance [1,2,6,7] for indepth investigations. Recall that the Friedrichs number c(L 1 , L 2 ) between the two subspaces L 1 , L 2 of X is defined as c(L 1 , L 2 ) = sup |(x 1 , x 2 )| : x k ∈ L k ∩ L ⊥ and x k ≤ 1 for k = 1, 2 , where L = L 1 ∩ L 2 .…”