The rate of convergence in the method of alternating projections
aa r X i v : . [ m a t h . F A ] J un THE RATE OF CONVERGENCE IN THE METHOD OFALTERNATING PROJECTIONS
CATALIN BADEA, SOPHIE GRIVAUX, AND VLADIMIR M ¨ULLER
Abstract.
A generalization of the cosine of the Friedrichs angle between two subspacesto a parameter associated to several closed subspaces of a Hilbert space is given. This pa-rameter is used to analyze the rate of convergence in the von Neumann-Halperin methodof cyclic alternating projections. General dichotomy theorems are proved, in the Hilbertor Banach space situation, providing conditions under which the alternative QUC/ASC(quick uniform convergence versus arbitrarily slow convergence) holds. Several meaningsfor ASC are proposed. Introduction
Throughout the paper H is a complex Hilbert space. For a closed linear subspace S of H we denote by S ⊥ its orthogonal complement in H , and by P S the orthogonal projectionof H onto S . In this paper N denotes a fixed positive integer greater or equal than 2.1A. The method of alternating projections.
It was proved by J. von Neumann [27,p. 475] that for two closed subspaces M and M of H , with intersection M = M ∩ M ,the following convergence result holds:(1.1) lim n →∞ k ( P M P M ) n ( x ) − P M ( x ) k = 0 ( x ∈ H ) . Using the notation T = P M P M , von Neumann’s result says that the iterates T n of T are strongly convergent to T ∞ = P M . The method of constructing the iterates of T by alternately projecting onto one subspace and then the other is called the method ofalternating projections . This algorithm, and its variations, occur in several fields, pure orapplied. We refer to [10, Chapter 9] as a source for more information.A generalization of von Neumann’s result to N closed subspaces M , . . . , M N withintersection M = M ∩ M · · · ∩ M N was proved by Halperin [16]: for each x ∈ H we have(1.2) lim n →∞ k ( P M N · · · P M P M ) n ( x ) − P M ( x ) k = 0 . The algorithm provided by Halperin’s result will be called in this paper the method ofcyclic alternating projections .A Banach space extension of Halperin’s result was proved by Bruck and Reich [9]: if X is a uniformly convex Banach space and P j , 1 ≤ j ≤ N , are N norm one projections in B ( X ), then the iterates of T = P N · · · P P are strongly convergent. The strong limit T ∞ is a projection of norm one onto the intersection of the ranges of P j . The same resultholds [3] if X is uniformly smooth and each projection P j is of norm one. It also holds The first two named authors have been partially supported by ANR Projet Blanc DYNOP. The thirdnamed author was supported by grant No. 201/09/0473 of GA ˇCR and IAA100190903 of GA AV.
Keywords : the method of alternating projections, Friedrichs angle, speed of convergence, spectral theory,uniformly convex Banach spaces.
To appear in : St. Petersburg Math. J. (translation from Russian of Algebra and Analysis) (2010),no. 5. Announced in C. R. Math. Acad. Sci. Paris (2010), no. 1-2, 53–56. [3] if X is a reflexive (complex) Banach space and each projection P j is hermitian (thatis, with real numerical range). We refer to [3] and the references therein for other Banachspace results of this type.An interesting extension of the method of cyclic alternating projections is the method ofrandom alternating projections . Let P j , 1 ≤ j ≤ N , be N orthogonal projections in B ( H ), M = ∩ Nj =1 Ran( P j ), and let ( i k ) k ≥ be a sequence from { , , . . . , N } (random samples).The method of random alternating projections asks about the convergence of the sequence( x n ) n ≥ given by x = x , x n = P i n x n − . It is an open problem to know whether ( x n ) n ≥ isalways convergent in the topology of H . The convergence of ( x n ) n ≥ in the weak topologyhas been proved by Amemiya and Ando [1]. If each j between 1 and N occurs infinitelymany times in the sequence of random samples, then the weak limit of ( x n ) n ≥ is P M x .We refer to [15, 30, 19] for results related to this problem.1B. The rate of convergence.
It is important for applications to know how fast thealgorithm given by the method of alternating projections, or its variations, converge. For N = 2 a quite complete description of the rate of convergence is known, it terms of thenotion of angle of subspaces. Definition 1.1. (Friedrichs angle) Let M and M be two closed subspaces of the Hilbertspace H with intersection M = M ∩ M . The Friedrichs angle between the subspaces M and M is defined to be the angle in [0 , π/
2] whose cosine is given by c ( M , M ) := sup {| h x, y i | : x ∈ M ∩ M ⊥ ∩ B H , y ∈ M ∩ M ⊥ ∩ B H } , where B H := { h ∈ H : k h k ≤ } is the unit ball of H . The minimal angle (or Dixmierangle) between the subspaces M and M is defined to be the angle in [0 , π/
2] whose cosineis given by c ( M , M ) := sup {| h x, y i | : x ∈ M ∩ B H , y ∈ M ∩ B H } . We note that c ( M , M ) = c ( M ∩ M ⊥ , M ∩ M ⊥ ), and that c ( M , M ) = 1 if M = { } .We also have c ( M , M ) = c ( M ⊥ , M ⊥ ). We refer to the survey paper [12] for moreinformation about different notions of angle between subspaces of infinite dimensionalHilbert spaces and their properties, and to [28, Lecture VIII] for different occurences ofthe Friedrichs angle in functional-theoretical problems.It was proved by Aronszajn [2] (upper bound) and by Kayalar and Weinert [17] (equal-ity) that k ( P M P M ) n − P M k = c ( M , M ) n − ( n ≥ . This formula shows that the sequence ( T n ) of iterates of T = P M P M converges uniformly to T ∞ = P M if and only if c ( M , M ) <
1, i.e., if the Friedrichs angle between M and M is positive. When this happens, the iterates of T = P M P M converge “quickly” (i.e.at the rate of a geometrical progression) to T ∞ = P M , in the following sense:(QUC) (quick uniform convergence) there exist C > α ∈ ]0 ,
1[ such that k T n − T ∞ k ≤ Cα n ( n ≥ . It is also known [11] that c ( M , M ) < M + M is closed, if and only if M ⊥ + M ⊥ is closed, if and only if ( M ∩ M ⊥ ) + ( M ∩ M ⊥ ) is closed.When M + M is not closed, we have strong, but not uniform convergence. It wasrecently proved by Bauschke, Deutsch and Hundal (see [5] for the history of this result)that, given any sequence of reals decreasing to zero, there exists a point in the spacewith the property that the convergence in the method of alternating projections (von HE METHOD OF ALTERNATING PROJECTIONS 3
Neumann’s theorem) is at least as slow as this sequence of reals. Thus the iterates ofthe product of two orthogonal projections converge quickly, or arbitrarily slowly. We callthis alternative the (QUC)/(ASC) dichotomy : one has quick uniform convergence orarbitrarily slow convergence. We shall consider several meanings of (ASC) in this paper.The results concerning the rate of convergence in Halperin’s theorem for N ≥ N = 2. We refer to [11, 13, 32], [10, Chapter9] and their references for several results concerning the rate of convergence in the methodof cyclic alternating projections. For instance, [13, Example 3.7] shows that for N ≥ c ( M i , M j ) between pairs of subspaces.1C. What this paper is about.
The main goal of the present paper is to discuss therate of convergence in Halperin’s theorem and to generalize some of the previous knownresults ( N = 2) to the case of several subspaces ( N ≥ T are strongly convergent. Several interpretations of (ASC) are proposed, and general di-chotomy theorems are obtained in the Hilbert or Banach space situation, depending onseveral spectral properties imposed upon the operator T . This implies at once the di-chotomy (QUC)/(ASC) in all above-mentioned generalizations of the method of alternat-ing projections. We also give a generalization of the Friedrichs angle to several subspaces, c ( M , · · · , M N ), and prove that condition (QUC) holds in Halperin’s theorem if and only if c ( M , · · · , M N ) <
1. Estimates for the error k ( P M N · · · P M P M ) n − P M k are given in thiscase and several statements equivalent to the condition c ( M , · · · , M N ) < P i k · · · P i of projections. Morespecific descriptions of these results, and information about how the paper is organized,are given below.1D. Conditions for arbitrarily slow convergence.
Several dichotomy theorems ofthe type quick uniform convergence versus arbitrarily slow convergence are proved in thispaper. The quick uniform condition is the condition (QUC) presented above. We shallconsider in Section 2 the following conditions for (ASC):(ASC1) (arbitrarily slow convergence, variant 1) for every ε > a n ) n ≥ of positive numbers such that lim n →∞ a n = 0, there exists a vector x ∈ X suchthat k x k < sup n a n + ε and k T n x − T ∞ x k ≥ a n for all n .(ASC2) (arbitrarily slow convergence, variant 2) for every sequence ( a n ) n ≥ of positivenumbers such that lim n →∞ a n = 0, there exists a dense subset of points x ∈ X such that k T n x − T ∞ x k ≥ a n for all but a finite number of n ’s.(ASC3) (arbitrarily slow convergence, variant 3) for every sequence ( a n ) n ≥ of positivenumbers such that lim n →∞ a n = 0, there exist two vectors x ∈ X and y ∈ X ∗ (thedual of X ) such that Re h T n x − T ∞ x, y i ≥ a n for all n ≥
1. Furthermore, if thereis a Banach space Y such that X is a (isometrical) subspace of Y ∗ , then the vector y can be chosen in Y ;(ASCH) (arbitrarily slow convergence, Hilbertian version) for every ε > a n ) n ≥ of positive numbers such that lim n →∞ a n = 0, there exists a vector x ∈ H such that k x k < sup n a n + ε and Re h T n x − T ∞ x, x i ≥ a n for all n ≥ H is supposed to be a complex Hilbert space.The dichotomy results of Section 2 are based upon general results about the existenceof large (weak) orbits of operators (see [26, 24, 25]). CATALIN BADEA, SOPHIE GRIVAUX, AND VLADIMIR M ¨ULLER
Let us recall here the main result of [25] concerning large weak orbits in the Banachspace setting:
Theorem 1.2 ([25]) . Let X be a Banach space which does not contain c , and T a boundedoperator on X such that belongs to the spectrum of T and k T n x k tends to zero as n tendsto infinity for every x ∈ X . Then for any sequence ( a n ) n ≥ such that a n tends to zeroas n tends to infinity, there exists a vector x ∈ X and a functional x ∗ ∈ X ∗ such that Re h T n x, x ∗ i ≥ a n for every n ≥ . We prove in Section 2 that if the iterates of T ∈ B ( X ) are strongly convergent, then onehas (QUC) or (ASC1). Also, if the iterates are strongly convergent, then the dichotomy(QUC)/(ASC2) holds. Condition (QUC) holds if and only if Ran( λI − T ), the range of λI − T , is closed for each λ in the unit circle ∂ D . In the case when T ∈ B ( X ) is a powerbounded, mean ergodic operator with spectrum σ ( T ) included in D ∪ { } , it is provedusing the Katznelson-Tzafriri theorem [20] that the iterates of T are strongly convergent.Therefore the previous dichotomies (QUC)/(ASC1) and (QUC)/(ASC2) apply. Moreover,the dichotomy (QUC)/(ASC3) holds whenever the Banach space X contains no isomorphiccopy of c . If X = H is a Hilbert space, then also the dichotomy (QUC)/(ASCH) holds.We prove here that the (QUC) condition holds if and only if Ran( I − T ) is closed. Ap-plications to products of projections of norm one are given. In particular, the dichotomy(QUC)/(ASC) holds, with several variants of (ASC), for the cases covered by the theoremsof von Neumann, Halperin, Bruck-Reich and those of [3].1E. A generalization of the Friedrichs angle.
In order to quantify the rate of con-vergence in the method of alternating projections, an extension of the cosine of Friedrichsangle to several subspaces ( M , . . . , M N ) will be given in Section 3. It is a parameter c ( M , . . . , M N ) which lies between 0 and 1, defined as follows: c ( M , · · · , M N ) = sup (cid:26) N − P j Dichotomy theorems.Theorem 2.1 ((QUC)/(ASC1) and (QUC)/(ASC2)) . Let X be a Banach space and let T ∈ B ( X ) be such that the sequence of iterates ( T n ) is strongly convergent to T ∞ ∈ B ( X ) .Then the following dichotomy holds : either (QUC), or (ASC1). The quick uniformconvergence (condition (QUC)) holds if and only if (2.1) for every λ ∈ ∂ D , Ran( λ − T ) is closed.In these statements, the condition (ASC1) can be replaced by (ASC2).Proof. Suppose that the sequence of iterates ( T n ) n ≥ is strongly convergent to T ∞ ∈ B ( X ).Then T is mean ergodic , i.e., the Ces`aro means ( I + T + · · · + T n − ) /n are stronglyconvergent. Therefore ([21, page 73]) the space X can be decomposed as the direct sumof the kernel of T − I and the closure of the range of the same operator, X = Ker( T − I ) ⊕ Ran( T − I ). Moreover, T ∞ is the projection onto Ker( T − I ) along Ran( T − I ).Notice also that T ∞ acts on the space Ker( T − I ) as the identity. With respect to thedecomposition X = Ker( T − I ) ⊕ Ran( T − I ) we can write T = (cid:18) T ∞ A (cid:19) for some A ∈ B (Ran( T − I )). It is not difficult to prove that for every λ ∈ C , the rangeRan( T − λI ) is closed if and only if Ran( A − λI ) is. The strong convergence of T n andthe Banach-Steinhaus theorem imply that T is power bounded , that is sup n ≥ k T n k < ∞ .Thus σ ( T ), the spectrum of T , is included in the closed unit disk. As σ ( T ) = { } ∪ σ ( A ),the same inclusion holds for σ ( A ). In particular, the spectral radius of A verifies r ( A ) ≤ Case (1). We have r ( A ) < 1. Notice that we have(2.2) T n − T ∞ = (cid:18) A n (cid:19) . Since r ( A ) < 1, there exist C > α ∈ ]0 , 1[ such that k A n k ≤ Cα n ( n ≥ . This estimate and (2.2) gives the quick uniform convergence condition (QUC). Case (2). We have r ( A ) = 1. Recall that k A n y k → n → ∞ , for each y ∈ Ran( T − I ).The conditions (ASC1) and (ASC2) follow now from [26, Thm 14, p. 333].Suppose that Case (1) is fulfilled, i.e. r ( A ) < 1. Then A − λ is invertible for every λ ∈ ∂ D . In particular, Ran( A − λ ) = Ran( T − I ) is closed for each λ ∈ ∂ D . ThusRan( T − λ ) is also closed, for each λ ∈ ∂ D .Suppose now that all subspaces Ran( T − λ ), λ ∈ ∂ D , and so all Ran( A − λ ), λ ∈ ∂ D ,are closed. Then r ( A ) < 1. Indeed, suppose that r ( A ) = 1 and let λ ∈ ∂ D ∩ σ ( A ) be apoint in the unimodular spectrum of A . Then the condition k A n y k → n → ∞ foreach y shows that λ cannot be an eigenvalue: if Ay = λy , then y = 0. Indeed, we have k y k = k λ − n A n y k = k A n y k → n → ∞ . Thus λ ∈ σ ( A ) \ σ p ( A ) and Ran( A − λ ) isclosed. Therefore A − λ is an upper semi-Fredholm operator. As A − λI is a limit ofinvertible operators A − n +1 n λI , the index ind( A − λI ) of A − λI is 0 . Hence A − λI isinvertible, a contradiction with the assumption that λ ∈ σ ( A ). Thus r ( A ) < (cid:3) CATALIN BADEA, SOPHIE GRIVAUX, AND VLADIMIR M ¨ULLER Remark 2.2. The following is a different argument for the last part of the proof, withoutthe use of Fredholm theory. As λ ∈ σ ( A ) \ σ p ( A ) and Ran( A − λ ) is closed, the operator A − λ is lower bounded, and thus λ is not in the approximate point spectrum of A . Asevery point in the boundary of the spectrum is in the approximate point spectrum, weobtain the desired contradiction. We refer the reader to [26] as a basic reference for thespectral theory of linear operators we are using in the present paper. Theorem 2.3 ((QUC)/(ASC3) and (QUC)/(ASCH)) . Let X be a Banach space andlet T ∈ B ( X ) be a power bounded, mean ergodic operator with spectrum σ ( T ) includedin D ∪ { } . Then the sequence of iterates T n is strongly convergent to a certain oper-ator T ∞ ∈ B ( X ) , and the dichotomies of Theorem 2.1 apply. Moreover, the dichotomy(QUC)/(ASC3) holds whenever X contains no isomorphic copy of c . If X = H is aHilbert space, then also the dichotomy (QUC)/(ASCH) holds.In all these statements, the quick uniform convergence condition (QUC) holds if andonly if (2.3) Ran( I − T ) is closed.Proof. Again, using the mean ergodicity and [21, page 73], the space X can be decomposedas the direct sum X = Ker( T − I ) ⊕ Ran( T − I ). According to the Katznelson-Tzafriritheorem [20], the power boundedness condition and the spectral condition σ ( T ) ⊂ D ∪ { } imply lim n →∞ k T n +1 − T n k = 0. This shows that the sequence of iterates ( T n ) of T converges strongly to 0 on the range of T − I . The same holds for the closure Ran( T − I ).As T acts like identity on Ker( T − I ), we get that ( T n ) n ≥ converges strongly to T ∞ , theprojection onto Ker( T − I ) along Ran( T − I ). Thus we can apply Theorem 2.1 to obtainthe dichotomies (QUC)/(ASC1) and (QUC)/(ASC2).Let us show that (QUC)/(ASC3) also holds if X contains no isomorphic copy of c .Using the notation of the proof of Theorem 2.1, if the condition (QUC) is not satisfied,then r ( A ) = 1 (Case (2) in the proof of Theorem 2.1). As σ ( T ) ⊂ D ∪ { } , the sameinclusion holds for the spectrum of A . Therefore 1 ∈ σ ( A ). Remark also that k A n y k → n → ∞ since ( T n ) converges strongly to T ∞ . We can now apply Theorem 1.2 providedthat X contains no isomorphic copy of c . To obtain the dichotomy (QUC)/(ASCH) if X = H is a Hilbert space, we use [24, Theorem 2] (see also [4, Theorem 1] for the case ofweak convergence). (cid:3) Applications to the method of alternating projections. We introduce firstsome notation, and recall for the convenience of the reader some Banach space terminology.Let N ≥ 2. Let X be a Banach space and let P , · · · , P N be N fixed projections ( P j = P j )acting on X . We denote by S = S ( P , · · · , P N ) the convex multiplicative semigroupgenerated by P , · · · , P N . Recall that this is the convex hull of the set of all productswith factors from P , · · · , P N , and that the convex hull of every multiplicative semigroupof operators is a semigroup.The space X is said to be uniformly convex if for every ε ∈ (0 , 1) there exists δ ∈ (0 , x and y , with k x k ≤ k y k ≤ k x + y k / > − δ implies k x − y k < ε . An (equivalent) definition of a uniformly smooth Banach space is thefollowing: X is uniformly smooth if its dual, X ∗ , is uniformly convex. We refer to [23] formore information.We call P ∈ B ( X ) a norm one projection (non-zero orthoprojection) if P = P and k P k = 1. A self-adjoint projection in a Hilbert space is called, as usual, an orthogonalprojection . Recall that an operator T on a Banach space X is called hermitian if its HE METHOD OF ALTERNATING PROJECTIONS 7 numerical range is real. This is equivalent to ask that k exp( itT ) k = 1 for every real t .Hermitian operators on Hilbert spaces coincide with the self-adjoint ones; see for instance[8] and the references therein. Theorem 2.4. Let N ≥ . Let X be a complex Banach space, and let P , · · · , P N be N projections on X . Let T be an operator in S ( P , · · · , P N ) . If one of the following conditionsbelow holds true, then the sequence of iterates of T converges strongly and every dichotomy(QUC)/(ASC1), (QUC)/(ASC2), (QUC)/(ASC3) and (QUC)/(ASCH) (if X = H is aHilbert space) applies: (i) the space X is uniformly convex and each P j , ≤ j ≤ N , is a norm one projection; (ii) the space X is uniformly smooth, and each P j , ≤ j ≤ N , is a norm one projec-tion; (iii) the space X is reflexive and for each j there exists r j with < r j < such that k P j − r j I k ≤ − r j . In particular, this holds if each P j is hermitian, ≤ j ≤ N .Proof. It was proved in [3] that in all three situations the spectrum of T ∈ S ( P , · · · , P N )is included in D ∪ { } and that the iterates of T are strongly convergent. We apply theabove dichotomy theorems. Notice that uniformly convex and uniformly smooth Banachspaces are reflexive, and that reflexive Banach spaces contain no copies of c . (cid:3) A generalization of Friedrichs angle for N subspaces As mentioned in the introduction, the rate of convergence in the method of alternatingprojections for two closed subspaces M and M is controlled by the Friedrichs angle c ( M , M ). We introduce and study in this section a generalization of Friedrichs angle for N subspaces.3A. Definition. In order to introduce our generalization of the cosine of the Friedrichs an-gle to several closed subspaces, we start by giving an equivalent definition of the Friedrichsangle c ( M , M ). Lemma 3.1. (a) Let M and M be two closed subspaces of H . Then c ( M , M ) = sup (cid:26) h m , m ik m k + k m k : m ∈ M , m ∈ M , ( m , m ) = (0 , (cid:27) . (b) Let M and M be two closed subspaces in H . Then c ( M , M ) = sup (cid:26) h m , m ik m k + k m k : m j ∈ M j ∩ M ⊥ , ( m , m ) = (0 , (cid:27) = sup (cid:26) h m , m i + h m , m ih m , m i + h m , m i : m j ∈ M j ∩ M ⊥ , ( m , m ) = (0 , (cid:27) . Proof. We give the proof only for the first equality of the second part. Denote by s thefirst supremum from the statement of part (b). For every admissible pair ( m , m ) with( m , m ) = (0 , 0) we have2 k m k + k m k Re h m , m i ≤ k m k · k m k Re h m , m i≤ | h m , m i |k m k · k m k≤ c ( M , M ) . CATALIN BADEA, SOPHIE GRIVAUX, AND VLADIMIR M ¨ULLER Therefore s ≤ c ( M , M ).For the reverse inequality, let ε > 0. Then there exist two elements x ∈ M ∩ M ⊥ and x ∈ M ∩ M ⊥ with k x k = 1 and k x k = 1 such that c ( M , M ) < | h x , x i | + ε . Let θ ∈ R be such that h x , x i = | h x , x i | e iθ , and set m = e − iθ x and m = x . Then m ∈ M ∩ M ⊥ , m ∈ M ∩ M ⊥ and k m k = 1, k m k = 1. We obtain s ≥ h m , m ik m k + k m k = Re D e − iθ x , x E = | h x , x i | > c ( M , M ) − ε. As ε is arbitrary, we obtain s = c ( M , M ). (cid:3) Definition 3.2. Let N ≥ 2. Let M , · · · , M N be N closed subspaces of H with intersection M = M ∩ · · · ∩ M N . The Dixmier number associated to ( M , · · · , M N ) is defined as c ( M , · · · , M N ) = sup (cid:26) N − P j Friedrichs number c ( M , · · · , M N ) associated to ( M , · · · , M N ) is defined as c ( M , · · · , M N ) = sup (cid:26) N − P j We found con-venient to introduce the following parameters, called the (reduced or not) configurationconstants, although they can be expressed in terms of the Dixmier and Friedrichs numbers(see Proposition 3.6, (f)). Definition 3.3. Let N ≥ 2. Let M , · · · , M N be N closed subspaces of H with intersection M = M ∩ · · · ∩ M N . The number κ ( M , · · · , M N ) = sup ( N k P Nj =1 m j k P Nj =1 k m j k : m j ∈ M j , k m k + · · · + k m N k = 0 ) is called the non-reduced configuration constant of ( M , · · · , M N ). The number κ ( M , · · · , M N ) = sup ( N k P Nj =1 m j k P Nj =1 k m j k : m j ∈ M j ∩ M ⊥ , k m k + · · · + k m N k = 0 ) is called the configuration constant of ( M , · · · , M N ).The configuration constant is related to the maximal possible norms of Gramian matri-ces. Recall that the Gramian matrix of an N -tuple of vectors ( v , · · · , v N ) is the N × N matrix G ( v , · · · , v N ) = [ h v i , v j i ] ≤ i,j ≤ N . HE METHOD OF ALTERNATING PROJECTIONS 9 Proposition 3.4. Let N ≥ . Let M , · · · , M N be N closed subspaces of H with inter-section M = M ∩ · · · ∩ M N . Then κ ( M , · · · , M N ) = sup (cid:26) N k G ( v , · · · , v N ) k : v j ∈ M j ∩ M ⊥ , k v j k = 1 , j = 1 , · · · , N (cid:27) . Proof. Let m j ∈ M j ∩ M ⊥ , j = 1 , · · · , N , with k m k + · · · + k m N k = 0. Set v j = m j k m j k if k m j k 6 = 0, or v j = 0 if m j = 0. Denote x = ( k m k , · · · , k m N k ) ∈ C N \ { } . We have h x , x i = x t x = 0 and1 N k P Nj =1 m j k P Nj =1 k m j k = 1 N DP Ni k m i k v i , P Nj k m j k v j EP Nj =1 k m j k = 1 N x t G ( v , · · · , v N ) xx t x . The conclusion follows by taking the supremum and noting that the Gramian matrix G ( v , · · · , v N ) is a Hermitian matrix. (cid:3) Consider the product Hilbert space H N which is the Hilbertian direct sum of N copiesof H , with scalar product h x , y i = h ( x , · · · , x N ) , ( y , · · · , y N ) i := N X j =1 h x j , y j i . We denote by C the Cartesian product C = M × · · · × M N ⊂ H N , and by D the diagonalsubset D = diag ( H ) = { ( y, . . . , y ) : y ∈ H } ⊂ H N . Recall that M = M ∩ · · · ∩ M N . Lemma 3.5. The projections onto C , D and C ∩ D are given by P C ( x , · · · , x N ) = ( P x , · · · , P N x N ) ,P D ( x , · · · , x N ) = (( x + · · · + x N ) /N, · · · , ( x + · · · + x N ) /N ) , and, respectively, by P C ∩ D ( x , · · · , x N ) = (( P M x + · · · + P M x N ) /N, · · · , ( P M x + · · · + P M x N ) /N ) , where ( x , · · · , x N ) ∈ H N .Proof. The formulae for P C and P D were proved in [29]. For the third one, we note firstthat C ∩ D = diag ( M ) = { ( m, · · · , m ) : m ∈ M } . Therefore k x − P C ∩ D x k = dist( x , diag ( M )) = inf { N X j =1 k x j − m k : m ∈ M } . We obtain k x − P C ∩ D x k = N X j =1 k x j − P M x j k + inf N X j =1 k P M x j − m k : m ∈ M . The infimum is realized when the gradient is zero, P Nj =1 ( m − P M x j ) = 0, that is when m = N − P Nj =1 P M x j . (cid:3) Proposition 3.6. Let N ≥ . Let M , · · · , M N be N closed subspaces of H with inter-section M = M ∩ · · · ∩ M N . Then (a) c ( M , · · · , M N ) = 1 if M = { } , while c ( M , · · · , M N ) = 0 if and only if thesubspaces ( M , · · · , M N ) are pairwise orthogonal; (b) c ( M , · · · , M N ) = c ( M ∩ M ⊥ , · · · , M N ∩ M ⊥ ) = c ( M ∩ M ⊥ , · · · , M N ∩ M ⊥ ) ,and thus c ( M , · · · , M N ) = c ( M , · · · , M N ) if M = { } ; (c) 0 ≤ c ( M , · · · , M N ) ≤ and ≤ c ( M , · · · , M N ) ≤ ; (d) N ≤ κ ( M , · · · , M N ) ≤ and N ≤ κ ( M , · · · , M N ) ≤ ; (e) κ ( M , · · · , M N ) = c ( C , D ) and κ ( M , · · · , M N ) = c ( C , D ) ; (f) c ( M , · · · , M N ) = NN − κ ( M , · · · , M N ) − N − = NN − c ( C , D ) − N − and similarstatements hold for c ( M , · · · , M N ) .Proof. We start by giving the proof of part (e). We have c ( C , D ) = sup (cid:26) | h ( m , . . . , m N ) , ( y, . . . , y ) i H N | N ( k m k + · · · + k m N k ) k y k : y ∈ H, y = 0 , m j ∈ M j ∩ M ⊥ , k m k + · · · + k m N k = 0 o = sup DP Nj =1 m j , y E N ( k m k + · · · + k m N k ) k y k : y ∈ H, y = 0 , m j ∈ M j ∩ M ⊥ , k m k + · · · + k m N k = 0 o = sup ( k P Nj =1 m j k N ( k m k + · · · + k m N k ) : y ∈ H, y = 0 , m j ∈ M j ∩ M ⊥ , k m k + · · · + k m N k = 0 o = κ ( M , . . . , M N ) . The proof of the equality κ ( M , · · · , M N ) = c ( C , D ) is similar.We prove now that c ( M , · · · , M N ) = NN − κ ( M , · · · , M N ) − N − . Indeed, we have c ( M , · · · , M N ) = sup (cid:26) N − P j For the lower bound κ ( M , · · · , M N ) ≥ /N , notice that we have, for m ∈ M \ { } , κ ( M , · · · , M N ) ≥ N kk m k k m k = 1 N . The inequalities for κ ( M , · · · , M N ) follow from κ ( M , · · · , M N ) = κ ( M ∩ M ⊥ , · · · , M N ∩ M ⊥ ) . Now (c) is a consequence of (f) and (d), while (b) and (a) are easy to prove. For thefirst equality in (b) notice that ∩ Nj =1 ( M j ∩ M ⊥ ) = { } . (cid:3) Proposition 3.7. Let N ≥ . Let M , · · · , M N be N closed subspaces of H with inter-section M = M ∩ · · · ∩ M N . Then κ ( M , · · · , M N ) = (cid:13)(cid:13)(cid:13)(cid:13) P + · · · + P N N − P M (cid:13)(cid:13)(cid:13)(cid:13) and c ( M , · · · , M N ) = NN − (cid:13)(cid:13)(cid:13)(cid:13) P + · · · + P N N − P M (cid:13)(cid:13)(cid:13)(cid:13) − N − . Proof. We have (see for instance [17]) c ( C , D ) = k P D P C − P C ∩ D k . Using Lemma 3.5, P D P C − P C ∩ D can be written as1 N · · · 11 1 · · · · · · P · · · 0. . . . . .0 0 · · · P N − N P M P M · · · P M P M · · · · · · P M ... ... P M P M · · · P M . Therefore P D P C − P C ∩ D = 1 N P − P M P − P M · · · P N − P M P − P M P − P M · · · P N − P M ... ... P − P M P − P M · · · P N − P M , and so κ := κ ( M , . . . , M N ) = c ( C , D ) = 1 N (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) P − P M P − P M · · · P N − P M P − P M P − P M · · · P N − P M ... ... P − P M P − P M · · · P N − P M (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) . We obtain that N κ is equal to (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) P − P M P − P M · · · P N − P M P − P M P − P M · · · P N − P M ... ... P − P M P − P M · · · P N − P M P − P M P − P M · · · P − P M P − P M P − P M · · · P − P M ... ... P N − P M P N − P M · · · P N − P M (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) . Therefore N κ = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Σ Σ · · · ΣΣ Σ · · · Σ... ...Σ Σ · · · Σ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) , where Σ := N X j =1 ( P j − P M ) . Since ( P j − P M ) = P j − P j P M − P M P j + P M = ( I − P M ) P j , we have Σ = ( I − P M ) N X j =1 P j . Let K be the matrix having all entries equal to Σ. One way to compute the norm of K isto note that, like every circulant matrix, K is unitarily equivalent to a diagonal matrix.Indeed, denote by F the N × N unitary matrix representing the discrete Fourier transform F = N − / [( ω jk )] ≤ j,k ≤ N − , where ω = exp( − iπ/N ) is a primitive N th root of unity.Then F ∗ KF = N Σ 0 · · · 00 0 · · · · · · . Therefore κ = 1 N k Σ k = k ( I − P M ) N X j =1 P j k = k N X j =1 P j ( I − P M ) k . Since P j P M = P M , this can be written as κ = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) P Nj =1 P j N − P M (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) . The proof is complete. (cid:3) The following definition is related to the minimum gap between two subspaces (see[18, p. 219 and Lemma 4.4]). See also the regularity (or boundedly linearly regularity)condition from [6], and the references therein. Definition 3.8. Let N ≥ 2. Let M , · · · , M N be N closed subspaces of H with intersection M = M ∩ · · · ∩ M N . The number ℓ ( M , . . . , M N ) = inf x/ ∈ M max ≤ j ≤ N dist( x, M j )dist( x, M )is called the inclination of ( M , · · · , M N ). Proposition 3.9. Let N ≥ . Let M , · · · , M N be N closed subspaces of H with inter-section M = M ∩ · · · ∩ M N . Then − ℓ ( M , . . . , M N ) ≤ c ( C , D ) = κ ( M , . . . , M N ) / ≤ − ℓ ( M , . . . , M N ) N and − NN − ℓ ( M , . . . , M N ) ≤ c ( M , . . . , M N ) ≤ − ℓ ( M , . . . , M N ) N − . HE METHOD OF ALTERNATING PROJECTIONS 13 In particular, ℓ ( M , . . . , M N ) = 0 if and only if c ( M , . . . , M N ) = 1 , if and only if κ ( M , . . . , M N ) = 1 .Proof. Denote ℓ = ℓ ( M , . . . , M N ). Let ε > 0. There exists x ∈ H with k x − P M x k =dist( x, M ) = 1 such that dist( x, M j ) < ℓ + ε for each j . Set u j = P j ( x − P M x ) = P j x − P M x ,where P j is the orthogonal projection onto M j (1 ≤ j ≤ N ). Then u j ∈ M j and k u j k ≤k x − P M x k = 1. Recall that C is the ℓ -direct sum C = M × · · · × M N ⊂ H N , while D isthe diagonal D = diag ( H ) = { ( y, . . . , y ) : y ∈ H } ⊂ H N . We have C ∩ D = diag ( M ),and so y = ( y , · · · , y N ) ∈ diag ( M ) ⊥ if and only if h y + · · · + y N , m i = 0 for every m ∈ M . Thus ( y , · · · , y N ) ∈ diag ( M ) ⊥ if and only if y + · · · + y N ∈ M ⊥ .Consider d = ( 1 √ N ( I − P M ) x, . . . , √ N ( I − P M ) x ) ∈ D ∩ diag ( M ) ⊥ and c = ( 1 √ N u , . . . , √ N u N ) ∈ C . For each m ∈ M we have * N X j =1 u j , m + = 1 √ N N X j =1 h P j x, m i − N h P M x, m i = 1 √ N N X j =1 h x, P j m i − N h x, m i = 0 . Therefore c ∈ C ∩ diag ( M ) ⊥ . We also have k d k = 1 and k c k = N ( k u k + · · · + k u N k ) ≤ 1. Thus c ( C , D ) ≥ |h c , d i| = 1 N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)* N X j =1 u j , x − P M x +(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ N Re * N X j =1 u j , x − P M x + . For a fixed j we have k x − P M x − u j k = k x − P j x k = dist( x, M j ) < ℓ + ε . Therefore2 Re h x − P M x, u j i = k x − P M x k + k u j k − k x − P M x − u j k > k u j k − ( ℓ + ε ) . We also have k u j k ≥ k x − P M x k − k x − P M x − u j k > − ( ℓ + ε ). We obtain c ( C , D ) ≥ N Re * N X j =1 u j , x − P M x + ≥ N N X j =1 (cid:0) k u j k − ( ℓ + ε ) (cid:1) ≥ N (cid:0) N + N (1 − ( ℓ + ε )) − N ( ℓ + ε ) (cid:1) = 12 (cid:0) − ( ℓ + ε )) − ( ℓ + ε ) (cid:1) . As this inequality is true for every ε > c ( C , D ) ≥ − ℓ + (1 − ℓ ) − ℓ. Denote c = c ( C , D ). Let ε > 0. There exist c ∈ C ∩ diag ( M ) ⊥ and d ∈ D ∩ diag ( M ) ⊥ with k c k = 1, k d k = 1 such that c < | h c , d i | + ε . Let θ ∈ R be such that h c , d i = e iθ | h c , d i | . Then k c − e iθ d k = 2 − e − iθ h c , d i ) = 2 − | h c , d i | ≤ − c − ε ) . Set c = ( m , . . . , m N ) ∈ C ∩ diag ( M ) ⊥ and e iθ d = ( y, . . . , y ) ∈ D ∩ diag ( M ) ⊥ . Then y ∈ M ⊥ and k m k + · · · + k m N k = 1 , k y k = 1 / √ N , and N X j =1 k m j − y k ≤ − c − ε ) . Let x = √ N y . Then x ∈ M ⊥ , dist( x, M ) = k x k = 1 and we havedist( x, M j ) ≤ k x − √ N m j k = N k m j − y k ≤ N (2 − c − ε )) . We finally obtain ℓ ≤ N (1 − c ), and so1 − ℓ ≤ c ( C , D ) ≤ − ℓ N . Using the equalities c ( C , D ) = κ ( M , . . . , M N ) / and κ ( M , . . . , M N ) = N − N c ( M , · · · , M N ) + 1 N we obtain N (1 − ℓ ) − N − ≤ c ( M , · · · , M N ) ≤ N (1 − ( ℓ / N ) ) − N − . Since 2 − ℓ ≤ 2, we can write N (1 − ℓ ) − N − − NN − ℓ (2 − ℓ ) ≥ − NN − ℓ. We also have N (1 − ( ℓ / N ) ) − N − − ℓ N − − ℓ N ) ≤ − ℓ N − . (cid:3) Characterising (ASC) for products of projections A qualitative result. When T is the product of N orthogonal projections, weknow from Theorem 2.4 that the dichotomy (QUC)/(ASC) holds, and that we have quickuniform convergence if and only if the range of T − I is closed. The following qualitativeresult gives a characterization of the (ASC) condition in terms of several parametersassociated to ( M , · · · , M N ), or spectral properties of T , or random products. We denoteby k · k e the essential norm and by σ e the essential spectrum . Theorem 4.1. Let N ≥ . Let M , · · · , M N be N closed subspaces of H with intersection M = M ∩ · · · ∩ M N . Denote P j the orthogonal projection onto M j , ≤ j ≤ N , and by P M the orthogonal projection onto M . Let T = P N P N − · · · P . The following assertionsare equivalent: (1) Ran( T − I ) is not closed; (1 ′ ) for every k ≥ N and every sequence of indices ( i k ) k ≥ such that { i , . . . , i k } = { , , . . . , N } , Ran( P i k · · · P i − I ) is not closed; (2) one of the conditions (ASC1), (ASC2), (ASC3), (ASCH) holds for T ; HE METHOD OF ALTERNATING PROJECTIONS 15 (2 ′ ) (ASCH) for random products: for every ε > , every sequence ( a n ) n ≥ of positivereals with lim n →∞ a n = 0 , and every sequence of indices ( i k ) k ≥ in { , , . . . , N } ,there exists x ∈ H with k x k < sup n a n + ε such that Re (cid:10) P i n P i n − · · · P i x − P M x, x (cid:11) > a n for each n ≥ ; (3) c ( M , · · · , M N ) = 1 . Equivalently, κ ( M , · · · , M N ) = 1 , or ℓ ( M , · · · , M N ) = 0 ; (4) for every ε > , every closed subspace K ⊂ M ⊥ of finite codimension (in M ⊥ ),there exists x ∈ K such that k x k = 1 and max { dist( x, M j ) : j = 1 , · · · , N } < ε ; (5) 1 ∈ σ ( T − P M ) ; (5 ′ ) for every k and every i , · · · , i k ∈ { , , · · · , N } we have ∈ σ ( P i k · · · P i − P M ) ; (6) k T − P M k = 1 ; (6 ′ ) for every k and every sequence of indices ( i k ) k ≥ , ≤ i k ≤ N , with { i , . . . , i k } = { , , . . . , N } we have k P i k · · · P i − P M k = 1 ; (7) k T − P M k e = 1 ; (7 ′ ) for every k and every i , · · · , i k ∈ { , , · · · , N } we have k P i k · · · P i − P M k e = 1 ; (8) 1 ∈ σ e ( T − P M ) ; (8 ′ ) for every k , every i , · · · , i k ∈ { , , · · · , N } we have ∈ σ e ( P i k · · · P i − P M ) ; (9) for every ε > , every closed subspace K ⊂ M ⊥ of finite codimension (in M ⊥ ),there exists x ∈ K such that k T x − x k ≤ ε ; (9 ′ ) for every ε > , every closed subspace K ⊂ M ⊥ of finite codimension (in M ⊥ ),there exists x ∈ K such that k P i k · · · P i x − x k ≤ ε for every k , every i , · · · , i k ∈{ , , · · · , N } ; (10) the sum of diag ( M ) ⊂ H N − and M ⊕ · · · ⊕ M N ⊂ H N − is not closed in H N − (and equivalent statements for diag ( M j ) ⊂ H N − , ≤ j ≤ N ); (11) M ⊥ + · · · + M ⊥ N is not closed in H . The conditions (1) , (2) , (5) , (6) , (7) , (8) and (9), most of them of spectral nature, areconditions about T = P N · · · P , while the corresponding conditions denoted with primesare analog conditions about random products P i N · · · P i . The conditions (3) , (4) , (10) and(11) are about the geometry of subspaces M j .Notice that we have the dichotomy (QUC)/(ASC) in all possible senses, and that (QUC)holds if and only if c ( M , . . . , M N ) < 1. A quantitative estimate reflecting the geometricconvergence of k T n − P M k to zero, in terms of the Friedrichs number, will be given afterthe proof of the theorem. Proof of Theorem 4.1. ”(1) ⇔ (2)“ The equivalence of (1) and (2) follows from Theorem2.4.”(1) ⇔ (5)“ The equivalence of (1) and (5) follows from the proof of Theorem 2.1(see also Remark 2.2). Notice that, with respect to the decomposition H = M ⊕ M ⊥ , wehave T = P M ⊕ A , where A = T | M ⊥ = T ( I − P M ) = T − P M .”(1) ⇒ (3)“ We prove this implication in a quantitative form. Denote γ = γ ( I − T ) = inf {k x − T x k : x ∈ H, dist( x, Ker( T − I ) = 1 } the reduced minimum modulus of T − I . Then Ran( T − I ) is closed if and only if γ > T y = y for y ∈ M . If T x = x , then k x k = k P N · · · P x k ≤ k P N − · · · P x k ≤ · · · ≤ k P x k ≤ k x k . We successively obtain P x = x , P x = x , . . . , P N x = x , and finally x ∈ M . ThusKer( T − I ) = M . Let ε > 0. There exists x ∈ H with k x − P M x k = dist( x, M ) = 1 such that k x − T x k ≤ γ + ε . We obtain1 = k x − P M x k ≥ k P ( x − P M x ) k = k P x − P M x k ≥ k P P x − P M x k≥ · · · ≥ k P N · · · P x − P M x k = k T x − P M x k≥ k x − P M x k − k x − T x k ≥ − γ − ε. We also have k ( I − P )( x − P M x ) k = k x − P M x k − k P x − P M x k ≤ − (1 − γ − ε ) = − ( γ + ε ) + 2( γ + ε ) ≤ γ + 2 ε. Thus dist( x, M ) = k x − P x k = k ( I − P )( x − P M x ) k ≤ (2 γ + 2 ε ) / .Let y = x − P M x ; then k y k = 1. For a fixed s between 1 and N we can write k P s · · · P y − P s +1 · · · P y k = k P s · · · P y k − k P s +1 · · · P y k ≤ k y k − k P s +1 · · · P x − P M x k ≤ − (1 − γ − ε ) ≤ γ + 2 ε. Thusdist( x, M j ) = dist( y, M j ) ≤ k y − P j · · · P y k≤ k y − P y k + k P y − P P y k + · · · + k P j − · · · P y − P j · · · P y k≤ j p (2 γ + 2 ε ) , for every j . Hence max ≤ j ≤ N dist( x, M j ) ≤ N p (2 γ + 2 ε ) and, as ε is arbitrary, ℓ := ℓ ( M , . . . , M N ) ≤ N p γ. We obtain N ℓ ( M , . . . , M N ) ≤ γ ( T − I ) . Therefore Ran( I − T ) not closed ( γ = 0)implies ℓ ( M , . . . , M N ) = 0.”(3) ⇒ (1 ′ )“ Let k ≥ N and let ( i k ) k ≥ be a sequence of indices with { i , . . . , i k } = { , , . . . , N } . This implies that Ker( I − P i k P i k − · · · P i ) = M . Let ℓ = ℓ ( M , . . . , M N )and let ε > 0. There exists x ∈ H with k x − P M x k = dist( x, M ) = 1 such thatmax j dist( x, M j ) < ℓ + ε . We have k x − P i x k = dist( x, M i ) < ℓ + ε and k P i P i x − P i x k = dist( P i x, M i ) ≤ k x − P i x k + dist( x, M i ) < ℓ + ε ) . Set x = x and x s = P i s P i s − · · · P i x for s ≥ 1. Suppose that(4.1) k x s − x s − k ≤ s − ( ℓ + ε )holds for 1 ≤ s ≤ r . Then k x r +1 − x r k = dist( P i r · · · P i x, M i r +1 ) ≤ k P i r · · · P i x − x k + dist( x, M i r +1 ) ≤ k x s − x s − k + k x s − − x s − k + · · · + k x − x k + dist( x, M r +1 ) ≤ (2 r − + 2 r − + · · · + 2 + 1 + 1)( ℓ + ε ) = 2 r ( ℓ + ε ) . HE METHOD OF ALTERNATING PROJECTIONS 17 Therefore (4.1) holds for every s , and we obtain k P i k P i k − · · · P i x − x k = k x k − x k≤ k x k − x k − k + k x k − − x k − k + · · · + k x − x k≤ (2 k − + 2 k − + · · · + 1)( ℓ + ε ) = (2 k − ℓ + ε ) . Thus γ ( P i k P i k − · · · P i − I ) ≤ (2 k − ℓ + ε ). Making ε → γ ( P i k P i k − · · · P i − I ) ≤ (2 k − ℓ. This shows that if ℓ = 0 or, equivalently, if c ( M , · · · , M N ) = 1, then the range of P i k P i k − · · · P i − I is not closed.The implication ”(1 ′ ) ⇒ (1)“ is clear. Note also that the above proof for k = N and i s = s implies that(4.2) 12 N ℓ ≤ γ ( T − I ) ≤ (2 N − ℓ. Here ℓ = ℓ ( M , · · · , M N ).”(1 ′ ) ⇒ (6 ′ )“ Note that k P i k · · · P i − P M k ≤ a := k P i k · · · P i − P M k < 1. We want to show that the range of I − P i k · · · P i is closed. Noticefirst that Ker( I − P i k · · · P i ) = M since { i , . . . , i k } = { , , . . . , N } . Let x ∈ H be suchthat dist( x, M ) = k x − P M x k = 1. We have k ( I − P i k · · · P i ) x k = k x − P M x + P M x − P i k · · · P i x k≥ − k P i k · · · P i x − P M x k = 1 − k ( P i k · · · P i − P M )( x − P M x ) k≥ − a. Therefore the reduced minimum modulus of I − P i k · · · P i ) verifies γ ( I − P i k · · · P i ) ≥ − k P i k · · · P i − P M k . In particular, Ran( I − P i k · · · P i ) is closed if a < ′ ) ⇒ (6)“ is easy. Lemma 4.2. Let x ∈ H , and set u j = P j · · · P x − P M x for j ≥ , u = x − P M x . Forevery j with ≤ j ≤ N we have k u j − − u j k ≤ k u j − k − k T x − P M x k ≤ k x − P M x k − k T x − P M x k . Proof. We can write k T x − P M x k = k u N k = k P N u N − k ≤ k u N − k ≤ · · · ≤ k u k = k x − P M x k . Therefore k u j − − u j k + k T x − P M x k = k u j − − P j u j − k + k P N · · · P j +1 P j u j − k ≤ k u j − − P j u j − k + k P j u j − k = k u j − k = k P j − · · · P ( x − P M x ) k ≤ k x − P M x k , completing the proof of the Lemma. (cid:3) We continue the proof of Theorem 4.1. ”(6) ⇒ (3)“ Let j between 1 and N . Using the Cauchy-Schwarz inequality andLemma 4.2 we obtaindist( x, M j ) ≤ k x − P j · · · P x k ≤ ( k x − P x k + k P x − P P x k + · · · + k P j − · · · P x − P j · · · P x k ) ≤ j (cid:0) k u − u k + k u − u k + · · · + k u j − − u j k (cid:1) ≤ j (cid:0) k x − P M x k − k T x − P M x k (cid:1) ≤ N (cid:0) k x − P M x k − k T x − P M x k (cid:1) . We get N (cid:0) k x − P M x k − k T x − P M x k (cid:1) ≥ max ≤ j ≤ N dist( x, M j ) ≥ ℓ k x − P M x k , which yields k T x − P M x k ≤ (cid:18) − ℓ N (cid:19) k x − P M x k . In particular(4.3) k T − P M k ≤ r − ℓ N . Therefore k T − P M k = 1 implies ℓ = 0, i.e., (6) implies (3).”(1) ⇒ (9)“ Let ε > 0. Let K ⊂ M ⊥ be a closed subspace of finite codimension in M ⊥ . With respect to the decomposition H = M ⊕ M ⊥ , the operator T has the followingmatrix decomposition T = (cid:18) I A (cid:19) . Since Ran( T − I ) is not closed, the range of the operator I − A , acting on M ⊥ , is not closed.This means that I − A ∈ B ( M ⊥ ) is not an upper semi-Fredholm operator, and thereforethere exists x ∈ K such that k x k = 1 and k x − Ax k ≤ ε . It follows that k x − T x k ≤ ε .”(9) ⇒ (4)“ Let x be as in (9). Then x ∈ K , k x k = 1, and k x − T x k ≤ ε . We have1 = k x k ≥ k P x k ≥ k P P x k ≥ . . . ≥ k T x k ≥ − ε. Set x s = P s P s − · · · P x for s ≥ x = x . Then x s ∈ M s ∩ M ⊥ for each s ≥ x s − − x s = ( I − P s ) x s − is orthogonal to x s . Hence k x s − − x s k = k x s − k − k x s k ≤ − (1 − ε ) < ε and k x s − − x s k ≤ √ ε , for each s . We obtaindist( x, M ) = k x − P x k = k x − x k ≤ √ ε. For s ≥ x, M s ) ≤ k x − P s P s − · · · P x k ; hencedist( x, M s ) ≤ k x − P x k + k P x − P P x k + · · · + k P s − · · · P x − P s P s − · · · P x k ≤ s √ ε. Therefore max { dist( x, M j ) : j = 1 , · · · , N } ≤ N √ ε. As ε > ⇒ (9 ′ )“ Suppose that (4) holds. Let ε > K ⊂ M ⊥ be a closed subspaceof finite codimension in M ⊥ . Then there exists x ∈ K such that dist( x, M ) = k x k = 1and max { dist( x, M j ) : j = 1 , · · · , N } ≤ ε . Let i , · · · , i k ∈ { , , . . . , N } . Set x = x , x s = P i s · · · P i x for s ≥ 1. Then x ∈ K and x s ∈ M ⊥ ∩ M i s for s ≥ HE METHOD OF ALTERNATING PROJECTIONS 19 We shall prove by induction the following two claims :(*) dist( x s , M j ) ≤ s ε ( j ≥ k x s − x k ≤ (2 s − ε ( s ≥ . Both claims are clearly true for s = 0. Suppose that both inequalities are true for some s ≥ 0. Then, using several times the induction hypothesis, we have k x s +1 − x k ≤ k x s +1 − x s k + k x s − x k = dist( x s , M i s +1 ) + k x s − x k≤ s ε + (2 s − ε = (2 s +1 − ε. For j ≥ x s +1 , M j ) ≤ k x s +1 − x k + dist( x , M j ) ≤ (2 s +1 − ε + ε = 2 s +1 ε. Thus both (*) and (**) are true ; in particular we have k P i k · · · P i x − x k = k x k − x k ≤ k ε. As ε > ′ ).”(9 ′ ) ⇒ (8 ′ )“ We have P s P s − · · · P − P M = P s P s − · · · P ( I − P M ), so the range ofthis operator is in M ⊥ . The assertion (9 ′ ) implies that 1 belongs to the essential spectrumof the restriction of P s P s − · · · P − P M to M ⊥ . Therefore 1 ∈ σ e ( P s P s − · · · P − P M ).The implication ”(8 ′ ) ⇒ (8)“ is clear.”(8) ⇒ (7) ⇒ (6)“ The statement (8) implies the following sequence of inequalitiesfor the essential spectral radius r e ( T − P M ) and the essential norm of T − P M :1 ≤ r e ( T − P M ) ≤ k T − P M k e ≤ k T − P M k = k P N · · · P ( I − P M ) k ≤ . Thus all inequalities are equalities.The proofs of implications ”(8 ′ ) ⇒ (7 ′ ) ⇒ (6 ′ )“ are similar. The implications ”(8 ′ ) ⇒ (5 ′ ) ⇒ (5)“ are clear.”(9 ′ ) ⇒ (2 ′ )“ Let A k be the operator P i k P i k − · · · P i − P M restricted to M ⊥ . Thecondition (9 ′ ) implies that 1 is in the boundary of the essential spectrum of the operator A k . According to [1], on the space M ⊥ the operators A k converge weakly to 0. Theassertion (2 ′ ) can be proved exactly as in [4, Theorem 1] by replacing there T n by A n .The implication ”(2 ′ ) ⇒ (2)“ is clear.”(10) ⇔ (3)“ We have ℓ ( M , · · · , M N ) = 0 if and only if c ( diag( M ) , M ⊕ · · · ⊕ M N ) = 1. The proof of this assertion is analogous to that of the first part of Proposition3.9. Therefore ℓ ( M , · · · , M N ) = 0, or equivalently c ( M , · · · , M N ) = 1, if and only ifdiag( M ) + M ⊕ · · · ⊕ M N is not closed in H N − .The implication ”(11) ⇔ (3)“ follows from [7]. The proof is complete. (cid:3) Quantitative statements. Some remarks concerning the proof of Theorem 4.1 arein order. Remark 4.3. The proof of Theorem 4.1 gives some quantitative information between theparameters involved. Some other estimates can be proved in a similar way. For instance, wepresent here the quantitative version of the implication ”(6 ′ ) ⇒ (3)“. Let k ≥ 1. Suppose i , . . . , i k ∈ { , , . . . , N } and { i , . . . , i k } = { , , . . . , N } . Denote ℓ = ℓ ( M , . . . , M N ) and a := k P i k · · · P i − P M k ≤ . Let ε > 0. There exists x ∈ H with k x k = 1 such that k P i k · · · P i x − P M x k > a − ε .Denote y = x − P M x and x s = P i s · · · P i x − P M x, x = x − P M x = y ( s ≥ . Clearly 1 = k x k = p k y k + k P M x k ≥ k y k = k x k ≥ k x k ≥ · · · ≥ k x k k > a − ε. Since x s − − x s = ( I − P i s ) x s − is orthogonal to M i s , and x s ∈ M i s , we have k x s − − x s k = k x s − k − k x s k ≤ k y k − ( a − ε ) . For each r ∈ { , . . . , k } we have k x − P M x − x r k ≤ k x − x k + k x − x k + · · · + k x r − − x r k≤ k p k y k − ( a − ε ) . Since { i , . . . , i k } = { , , . . . , N } , for each j ∈ { , , . . . , N } we have { x , . . . , x k }∩ M j = ∅ .Thereforedist( x, M j ) ≤ max {k x − P M x − x r k : 1 ≤ r ≤ k } ≤ k p k y k − ( a − ε ) and k y k ℓ ≤ max { dist( x, M j ) : 1 ≤ j ≤ k } ≤ k (cid:0) k y k − ( a − ε ) (cid:1) . Hence k ( a − ε ) ≤ ( k − ℓ ) k y k ≤ ( k − ℓ ). As this is satisfied for every ε > 0, we obtain k P i k · · · P i − P M k = a ≤ (1 − ℓ /k ) / . Theorem 4.4. Let N ≥ . Let H be a complex Hilbert space. Let M , · · · , M N be N closedsubspaces of H with intersection M = M ∩ M · · · ∩ M N . Let P j = P M j , ≤ j ≤ N , and P M be the corresponding orthogonal projections. Denote T = P N · · · P .(i) Suppose that c := c ( M , · · · , M N ) < . Then ( T n ) n ≥ is uniformly convergent to P M , with k T n − P M k ≤ − (cid:18) − c N (cid:19) ! n/ ( n ≥ . (ii) Suppose that c := c ( M , · · · , M N ) = 1 . Then ( T n ) n ≥ is strongly convergent to P M and we have (ASC), in all possible meanings of this paper.Proof. We have to prove only the estimate in Part ( i ). Suppose that c := c ( M , · · · , M N ) < 1. Denote f M j = M j ∩ M ⊥ and Q j = P f M j for 1 ≤ j ≤ N . Then the intersection of f M j ,1 ≤ j ≤ N , is { } and, according to Proposition 3.6, (b), we have c ( f M , . . . , g M N ) = c < T n − P M = ( Q N Q N − · · · Q ) n for each n ≥ ⇒ (3)“ of Theorem 4.1, to Q j .We obtain k Q N Q N − · · · Q k ≤ q − ℓ N , where now ℓ = ℓ ( f M , . . . , g M N ). Accordingto Proposition 3.9, applied for the subspaces f M j , we have 1 − NN − ℓ ≤ c . This implies ℓ ≥ (cid:0) N − N (cid:1) (1 − c ) ≥ (1 − c ) . Therefore k T n − P M k = k ( Q N Q N − · · · Q ) n k ≤ k ( Q N Q N − · · · Q ) k n ≤ (cid:18) − ℓ N (cid:19) n ≤ (cid:18) − (cid:18) (1 − c ) N (cid:19)(cid:19) n , which implies ( i ). (cid:3) HE METHOD OF ALTERNATING PROJECTIONS 21 Comparison with other estimates. Let M , . . . , M N be N closed subspaces of H , with intersection M = M ∩ · · · ∩ M N . Denote c ij = c ( M i ∩ M ⊥ , M j ∩ M ⊥ ) for1 ≤ i, j ≤ N .It was proved in [13, Theorem 2.1] that k ( P N . . . P ) n − P M k ≤ c n − N c n . . . c nN − ,N . In particular, we have quick uniform convergence whenever one of the cosine c i,i +1 of theDixmier angles is strictly less than one.Moreover, for any sequence i , . . . i N of integers such that { i , . . . , i N } = { , . . . , N } ,Theorem 4.1 shows that we have (QUC) for ( P N . . . P ) n if and only if we have (QUC) for( P i N . . . P i ) n . Hence we have (QUC) for T n = ( P N . . . P ) n as soon as there exist integers i = j such that c ij = c ( M i ∩ M ⊥ , M j ∩ M ⊥ ) < 1. The following example shows that thissufficient condition for (QUC) is by far stronger than the condition c ( M , . . . , M N ) < Example 4.5. Let ( e n ) n ≥ be an orthonormal basis of H , and let M , M and M be thefollowing closed subspaces of H : M = span[ e n ; n ≥ M = span[ e , e n +1 ; n ≥ M = span[ e , e , e n +2 ; n ≥ M ∩ M = span[ e ], M ∩ M = span[ e ], M ∩ M = span[ e ] and M ∩ M ∩ M = { } . We obtain c ij = c ( M i , M j ) = 1 forany i and j . But c ( M , M , M ) < 1: indeed if x = P n ≥ x n e n , then a straightforwardcomputation shows that || ( P + P + P ) || = , so that c ( M , M , M ) = .The following proposition shows, even in a quantitative way, that the sufficient condition c ( M ∩ · · · ∩ M j − , M j ) < 1, for each j , reminiscent of [31, Theorem 2.2] and [13, Theorem2.7], implies that c ( M , . . . , M N ) < Proposition 4.6. Let N ≥ . Let M , · · · , M N be N closed subspaces of H with inter-section M = M ∩ · · · ∩ M N . Denote c j = c ( M ∩ · · · ∩ M j − , M j ) for j between and N .Then c ( M , . . . , M N ) ≤ − N − N Y j =2 − r c j + 12 ! ≤ − N − N − N Y j =2 (1 − c j ) . In particular, c ( M , . . . , M N ) < if each c j < , ≤ j ≤ N .Proof. The estimates are clear if one of the c j ’s is one. Suppose c j < j . Denote ℓ j = c ( M ∩ · · · ∩ M j − , M j ) > j between 2 and N . It follows from the proof of [6,Theorem 5.11] that ℓ ( M , . . . , M N ) ≥ ℓ ℓ · · · ℓ N . The proof of Proposition 3.9 for N = 2,and two given subspaces S and S , yields1 − ℓ ( S , S ) ≤ p κ ( S , S ) = r c ( S , S ) + 12 . This implies that ℓ ( S , S ) ≥ − r c ( S , S ) + 12 ≥ − c ( S , S )4 . Using Proposition 3.9 we obtain c ( M , . . . , M N ) ≤ − ℓ ( M , . . . , M N ) N − ≤ − ℓ ℓ · · · ℓ N N − ≤ − N − N Y j =2 − r c j + 12 ! ≤ − N − N − N Y j =2 (1 − c j ) , which completes the proof. 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Laboratoire Paul Painlev´e, Universit´e Lille 1, CNRS UMR 8524, F-59655 Villeneuved’Ascq, France E-mail address : [email protected] Laboratoire Paul Painlev´e, CNRS UMR 8524, Universit´e Lille 1, F-59655 Villeneuved’Ascq, France E-mail address : [email protected] Institute of Mathematics AV CR, Zitna 25, 115 67 Prague 1, Czech Republic E-mail address ::