THE RATE OF CONVERGENCE IN THE METHOD OF ALTERNATING PROJECTIONS

ALTERNATING PROJECTIONS

CATALIN BADEA, SOPHIE GRIVAUX, AND VLADIMIR M ¨ULLER

Abstract. A generalization of the cosine of the Friedrichs angle between two subspaces to 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 method of cyclic alternating projections. General dichotomy theorems are proved, in the Hilbert or Banach space situation, providing conditions under which the alternative QUC/ASC (quick uniform convergence versus arbitrarily slow convergence) holds. Several meanings for ASC are proposed.

1. Introduction

Throughout the paper H is a complex Hilbert space. For a closed linear subspaceS of H we denote byS^⊥its orthogonal complement inH, and byPS the orthogonal projection of H onto S. In this paperN 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 M1 andM2 of H, with intersectionM =M1∩M2, the following convergence result holds:

(1.1) lim

n→∞k(P_M2PM1)ⁿ(x)−PM(x)k= 0 (x∈H).

Using the notation T = PM2PM1, von Neumann’s result says that the iterates Tⁿ 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 of alternating projections. This algorithm, and its variations, occur in several fields, pure or applied. We refer to [10, Chapter 9] as a source for more information.

A generalization of von Neumann’s result to N closed subspaces M1, . . . , MN with intersectionM =M₁∩M₂· · · ∩M_N was proved by Halperin [16]: for eachx∈H we have

(1.2) lim

n→∞k(P_MN· · ·P_M2P_M1)ⁿ(x)−P_M(x)k= 0.

The algorithm provided by Halperin’s result will be called in this paper the method of cyclic 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 Pj, 1≤ j ≤N, are N norm one projections in B(X), then the iterates of T =P_N· · ·P₂P₁ are strongly convergent. The strong limit T^∞is a projection of norm one onto the intersection of the ranges of Pj. The same result holds [3] if X is uniformly smooth and each projection P_j is of norm one. It also holds [3] if X is a reflexive (complex) Banach space and each projection Pj is hermitian (that is, with real numerical range). We refer to [3] and the references therein for other Banach space results of this type.

The first two named authors have been partially supported by ANR Projet Blanc DYNOP. The third named author was supported by grant No. 201/09/0473 of GA ˇCR and IAA100190903 of GA AV.

1

An interesting extension of the method of cyclic alternating projections is themethod of random alternating projections. Let Pj, 1≤j ≤N, beN orthogonal projections inB(H), M = ∩^N_j=1Ran(Pj), and let (i_k)k≥1 be a sequence from {1,2, . . . , N} (random samples).

The method of random alternating projections asks about the convergence of the sequence (x_n)n≥0 given byx₀=x,x_n=P_inxn−1. It is an open problem to know whether (x_n)n≥0 is always convergent in the topology ofH. The convergence of (xn)n≥0 in theweak topology has been proved by Amemiya and Ando [1]. If each j between 1 and N occurs infinitely many times in the sequence of random samples, then the weak limit of (x_n)n≥0 is P_Mx.

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 the algorithm 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 the notion of angle of subspaces.

Definition 1.1. (Friedrichs angle) LetM₁ andM₂ be two closed subspaces of the Hilbert spaceH with intersectionM =M1∩M2. TheFriedrichs angle between the subspacesM1

and M₂ is defined to be the angle in [0, π/2] whose cosine is given by

c(M1, M2) := sup{| hx, yi |:x∈M1∩M^⊥∩BH, y ∈M2∩M^⊥∩BH},

where B_H := {h ∈ H :khk ≤ 1} is the unit ball of H. The minimal angle (or Dixmier angle) between the subspacesM₁andM₂ is defined to be the angle in [0, π/2] whose cosine is given by

c₀(M₁, M₂) := sup{| hx, yi |:x∈M₁∩B_H, y∈M₂∩B_H}.

We note thatc(M1, M2) =c0(M1∩M^⊥, M2∩M^⊥), and thatc0(M1, M2) = 1 ifM 6={0}.

We also have c(M1, M2) = c(M₁^⊥, M₂^⊥). We refer to the survey paper [12] for more information about different notions of angle between subspaces of infinite dimensional Hilbert spaces and their properties, and to [28, Lecture VIII] for different occurences of the 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_M2PM1)ⁿ−PMk=c(M1, M2)²ⁿ⁻¹ (n≥1).

This formula shows that the sequence (Tⁿ) of iterates ofT =PM2PM1 convergesuniformly to T^∞ = P_M if and only if c(M₁, M₂) < 1, i.e., if the Friedrichs angle between M₁ and M₂ is positive. When this happens, the iterates of T =P_M2P_M1 converge “quickly” (i.e.

at the rate of a geometrical progression) to T^∞=PM, in the following sense:

(QUC) (quick uniform convergence) there exist C >0 andα∈]0,1[ such that kTⁿ−T^∞k ≤Cαⁿ (n≥1).

It is also known [11] thatc(M1, M2)<1 if and only ifM1+M2 is closed, if and only if M₁^⊥+M₂^⊥ is closed, if and only if (M1∩M^⊥) + (M2∩M^⊥) is closed.

When M₁ +M₂ is not closed, we have strong, but not uniform convergence. It was recently 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 space with the property that the convergence in the method of alternating projections (von Neumann’s theorem) is at least as slow as this sequence of reals. Thus the iterates of the product of two orthogonal projections converge quickly, or arbitrarily slowly. We call

this alternative the (QUC)/(ASC) dichotomy : one has quick uniform convergence or arbitrarily slow convergence. We shall consider several meanings of (ASC) in this paper.

The results concerning the rate of convergence in Halperin’s theorem forN ≥3 are not as complete as the results described above forN = 2. We refer to [11, 13, 32], [10, Chapter 9] and their references for several results concerning the rate of convergence in the method of cyclic alternating projections. For instance, [13, Example 3.7] shows that for N ≥ 3 the error bound for the method of cyclic alternating projections is not a function of the various Friedrichs angles 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 the rate of convergence in Halperin’s theorem and to generalize some of the previous known results (N = 2) to the case of several subspaces (N ≥3). We show by operator-theoretical methods that the (QUC)/(ASC) dichotomy always holds as soon as the iterates of 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 on several 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(M₁,· · ·, M_N), and prove that condition (QUC) holds in Halperin’s theorem if and only if c(M1,· · ·, MN)<1. Estimates for the errork(P_MN· · ·PM2PM1)ⁿ−PMkare given in this case and several statements equivalent to the conditionc(M1,· · ·, M_N)<1 are obtained.

Some of them are expressed in terms of random products P_ik· · ·P_i1 of projections. More specific descriptions of these results, and information about how the paper is organized, are given below.

1D. Conditions for arbitrarily slow convergence. Several dichotomy theorems of the type quick uniform convergenceversus arbitrarily slow convergence are proved in this paper. The quick uniform condition is the condition (QUC) presented above. We shall consider in Section 2 the following conditions for (ASC):

(ASC1) (arbitrarily slow convergence, variant 1) for everyε >0 and every sequence (a_n)n≥1

of positive numbers such that limn→∞an = 0, there exists a vector x ∈ X such thatkxk<sup_na_n+ε andkTⁿx−T^∞xk ≥a_n for all n.

(ASC2) (arbitrarily slow convergence, variant 2) for every sequence (an)n≥1 of positive numbers such that limn→∞a_n = 0, there exists a dense subset of points x ∈ X such thatkTⁿx−T^∞xk ≥an for all but afinite number of n’s.

(ASC3) (arbitrarily slow convergence, variant 3) for every sequence (a_n)n≥1 of positive numbers such that limn→∞a_n= 0, there exist two vectors x∈X andy∈X^∗ (the dual ofX) such that RehTⁿx−T^∞x, yi ≥an for alln≥1. Furthermore, if there is a Banach spaceY such thatX is a (isometrical) subspace ofY^∗, then the vector y can be chosen inY;

(ASCH) (arbitrarily slow convergence, Hilbertian version) for every ε > 0 and every se- quence (an)n≥1 of positive numbers such that limn→∞an= 0, there exists a vector x ∈H such thatkxk < sup_na_n+εand RehTⁿx−T^∞x, xi ≥ a_n for all n≥ 1 . HereH is supposed to be a complex Hilbert space.

The dichotomy results of Section 2 are based upon general results about the existence of large (weak) orbits of operators (see [26, 24, 25]).

Let us recall here the main result of [25] concerning large weak orbits in the Banach space setting:

Theorem 1.2([25]). LetXbe a Banach space which does not containc₀, andT a bounded operator onX such that1belongs to the spectrum ofT andkTⁿxktends to zero asntends to infinity for every x ∈ X. Then for any sequence (an)n≥0 such that an tends to zero as n tends to infinity, there exists a vector x ∈ X and a functional x^∗ ∈ X^∗ such that RehTⁿx, x^∗i ≥an for every n≥0.

We prove in Section 2 that if the iterates ofT ∈ B(X) are strongly convergent, then one has (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 λin the unit circle∂D. In the case when T ∈ B(X) is a power bounded, mean ergodic operator with spectrum σ(T) included in D∪ {1}, it is proved using the Katznelson-Tzafriri theorem [20] that the iterates ofT are strongly convergent.

Therefore the previous dichotomies (QUC)/(ASC1) and (QUC)/(ASC2) apply. Moreover, the dichotomy (QUC)/(ASC3) holds whenever the Banach spaceXcontains no isomorphic copy ofc₀. IfX =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 theorems of 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 Friedrichs angle to several subspaces (M₁, . . . , M_N) will be given in Section 3. It is a parameter c(M1, . . . , MN) which lies between 0 and 1, defined as follows:

c(M1,· · ·, MN) = sup 2

N −1 P

j<kRehm_j, m_ki km₁k²+· · ·+km_Nk² :

m_j ∈M_j∩M^⊥,km₁k²+· · ·+km_Nk² 6= 0o

= sup ( 1

N −1 P

j6=khm_j, mki PN

j=1hm_j, mji :

m_j ∈M_j∩M^⊥,km₁k²+· · ·+km_Nk² 6= 0o .

The fact that this definition coincides with the classical one for two subspaces will be proved in Lemma 3.1. For pairwise orthogonal N subspaces (the “angle” is π/2 in this case) we havec(M₁, . . . , M_N) = 0, while the other extremal casec(M₁, . . . , M_N) = 1 cor- responds to the case of arbitrarily slow convergence in the method of cyclic alternating projections for N subspaces (the “angle” is zero). Other related quantities are consid- ered: theconfiguration constant κ(M1, . . . , MN), the inclination `(M1, . . . , MN), and the Friedrichs angle between the cartesian product C = M₁ × · · · × M_N ⊂ H^N and the

”diagonal subspace”D= diag (H) ={(y, . . . , y) :y∈H} ⊂H^N.

In Section 4 we characterize in several ways when the dichotomy (QUC)/(ASC) arises.

The characterizations are in terms of geometric properties of (M1,· · · , MN), of spectral properties of T, or of random productsP_ik· · ·P_i1. We give an estimate for the geometric convergence ofkTⁿ−PMk to zero whenc(M1, . . . , MN)<1.

2. General dichotomy theorems and applications 2A. 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(Tⁿ) is strongly convergent toT^∞∈ B(X).

Then the following dichotomy holds : either (QUC), or (ASC1). The quick uniform convergence (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 (Tⁿ)n≥0is strongly convergent toT^∞∈ B(X).

Then T is mean ergodic, i.e., the Ces`aro means (I +T +· · ·+Tⁿ⁻¹)/n are strongly convergent. Therefore ([21, page 73]) the space X can be decomposed as the direct sum of 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 the decomposition X= Ker(T −I)⊕Ran(T−I) we can write

T =

T^∞ 0

0 A

for some A ∈ B(Ran(T −I)). It is not difficult to prove that for everyλ∈ C, the range Ran(T −λI) is closed if and only if Ran(A−λI) is. The strong convergence of Tⁿ and the Banach-Steinhaus theorem imply that T ispower bounded, that is sup_n≥1kTⁿk<∞.

Thusσ(T), the spectrum ofT, is included in the closed unit disk. Asσ(T) ={1} ∪σ(A), the same inclusion holds forσ(A). In particular, the spectral radius ofAverifiesr(A)≤1.

We distinguish two cases.

Case (1). We have r(A)<1. Notice that we have

(2.2) Tⁿ−T^∞=

0 0 0 Aⁿ

.

Since r(A)<1, there exist C >0 andα∈]0,1[ such that kAⁿk ≤Cαⁿ (n≥1).

This estimate and (2.2) gives the quick uniform convergence condition (QUC).

Case (2). We haver(A) = 1. Recall thatkAⁿyk →0 asn→ ∞, for eachy∈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. Thus Ran(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 a point in the unimodular spectrum of A. Then the condition kAⁿyk → 0 as n → ∞ for each y shows that λ cannot be an eigenvalue: if Ay = λy, then y = 0. Indeed, we have kyk = kλ⁻ⁿAⁿyk = kAⁿyk → 0 as n → ∞. Thus λ ∈ σ(A)\σ_p(A) and Ran(A−λ) is closed. Therefore A−λ is an upper semi-Fredholm operator. As A−λI is a limit of invertible operators A− ⁿ⁺¹_n λI, the index ind(A−λI) of A−λI is 0.Hence A−λI is invertible, a contradiction with the assumption that λ∈σ(A). Thusr(A)<1.

Remark 2.2. The following is a different argument for the last part of the proof, without the use of Fredholm theory. As λ∈σ(A)\σp(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. As

every point in the boundary of the spectrum is in the approximate point spectrum, we obtain the desired contradiction. We refer the reader to [26] as a basic reference for the spectral 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 and let T ∈ B(X) be a power bounded, mean ergodic operator with spectrum σ(T) included in D∪ {1}. Then the sequence of iterates Tⁿ 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 c0. If X = H is a Hilbert space, then also the dichotomy (QUC)/(ASCH) holds.

In all these statements, the quick uniform convergence condition (QUC) holds if and only if

(2.3) Ran(I−T) is closed.

Proof. Again, using the mean ergodicity and [21, page 73], the spaceXcan be decomposed as the direct sum X = Ker(T −I)⊕Ran(T−I). According to the Katznelson-Tzafriri theorem [20], the power boundedness condition and the spectral conditionσ(T)⊂D∪ {1}

imply limn→∞kTⁿ⁺¹ −Tⁿk = 0. This shows that the sequence of iterates (Tⁿ) of T converges strongly to 0 on the range ofT−I. The same holds for the closure Ran(T−I).

AsT acts like identity on Ker(T−I), we get that (Tⁿ)n≥0 converges strongly toT^∞, the projection onto Ker(T −I) along Ran(T −I). Thus we can apply Theorem 2.1 to obtain the dichotomies (QUC)/(ASC1) and (QUC)/(ASC2).

Let us show that (QUC)/(ASC3) also holds if X contains no isomorphic copy of c₀. 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∪ {1}, the same inclusion holds for the spectrum of A. Therefore 1∈σ(A). Remark also thatkAⁿyk →0 asn→ ∞since (Tⁿ) converges strongly toT^∞. We can now apply Theorem 1.2 provided that X contains no isomorphic copy of c0. 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 of

weak convergence).

2B. Applications to the method of alternating projections. We introduce first some notation, and recall for the convenience of the reader some Banach space terminology.

LetN ≥2. LetXbe a Banach space and letP₁,· · ·, P_N beN fixed projections (P_j² =P_j) acting on X. We denote by S = S(P₁,· · · , P_N) the convex multiplicative semigroup generated by P₁,· · · , P_N. Recall that this is the convex hull of the set of all products with factors fromP1,· · · , PN, and that the convex hull of every multiplicative semigroup of operators is a semigroup.

The spaceX is said to be uniformly convex if for every ε∈(0,1) there exists δ∈(0,1) such that for any two vectors, x and y, with kxk ≤ 1 and kyk ≤ 1, kx+yk/2 > 1−δ implieskx−yk< ε. An (equivalent) definition of auniformly smooth Banach space is the following: X is uniformly smooth if its dual, X^∗, is uniformly convex. We refer to [23] for more information.

We call P ∈ B(X) a norm one projection (non-zero orthoprojection) if P² = P and kPk = 1. A self-adjoint projection in a Hilbert space is called, as usual, an orthogonal projection. Recall that an operator T on a Banach space X is called hermitian if its numerical range is real. This is equivalent to ask that kexp(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 ≥2. Let X be a complex Banach space, and let P₁,· · ·, P_N be N projections onX. LetT be an operator inS(P₁,· · ·, PN). If one of the following conditions below holds true, then the sequence of iterates ofT converges strongly and every dichotomy (QUC)/(ASC1), (QUC)/(ASC2), (QUC)/(ASC3) and (QUC)/(ASCH) (if X = H is a Hilbert space) applies:

(i) the spaceX is uniformly convex and eachPj,1≤j ≤N, is a norm one projection;

(ii) the space X is uniformly smooth, and each P_j, 1≤j≤N, is a norm one projec- tion;

(iii) the space X is reflexive and for each j there exists rj with 0 < rj <1 such that kP_j−r_jIk ≤1−r_j. In particular, this holds if each P_j is hermitian, 1≤j≤N. Proof. It was proved in [3] that in all three situations the spectrum ofT ∈ S(P₁,· · · , P_N) is included in D∪ {1} and that the iterates ofT are strongly convergent. We apply the above dichotomy theorems. Notice that uniformly convex and uniformly smooth Banach spaces are reflexive, and that reflexive Banach spaces contain no copies of c0.

3. A generalization of Friedrichs angle for N subspaces

As mentioned in the introduction, the rate of convergence in the method of alternating projections for two closed subspaces M₁ and M₂ is controlled by the Friedrichs angle c(M1, M2). 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 Friedrichs angle c(M₁, M₂).

Lemma 3.1. (a) Let M₁ and M₂ be two closed subspaces of H. Then c₀(M₁, M₂) = sup

2 Rehm₁, m₂i

km₁k²+km₂k² :m₁ ∈M₁, m₂∈M₂,(m₁, m₂)6= (0,0)

.

(b) Let M1 andM2 be two closed subspaces in H. Then c(M1, M2) = sup

2 Rehm₁, m2i

km₁k²+km₂k² :mj ∈Mj∩M^⊥,(m1, m2)6= (0,0)

= sup

hm₁, m₂i+hm₂, m₁i

hm₁, m1i+hm₂, m2i :m_j ∈M_j∩M^⊥,(m₁, m₂)6= (0,0)

.

Proof. We give the proof only for the first equality of the second part. Denote by s the first supremum from the statement of part (b). For every admissible pair (m1, m2) with (m₁, m₂)6= (0,0) we have

2

km₁k²+km₂k²Rehm₁, m2i ≤ 1

km₁k · km₂kRehm₁, m2i

≤ | hm₁, m2i | km₁k · km₂k

≤ c(M1, M2).

Therefores≤c(M1, M2).

For the reverse inequality, letε >0. Then there exist two elementsx1∈M1∩M^⊥and x₂ ∈ M₂ ∩M^⊥ with kx₁k = 1 and kx₂k = 1 such that c(M₁, M₂) <| hx₁, x₂i |+ε. Let

θ ∈ R be such that hx₁, x₂i = | hx₁, x₂i |e^iθ, and set m₁ = e^−iθx₁ and m₂ = x₂. Then m1 ∈M1∩M^⊥,m2 ∈M2∩M^⊥ and km₁k= 1, km₂k= 1. We obtain

s≥ 2 Rehm₁, m2i km₁k²+km₂k² = Re

D

e^−iθx1, x2

E

=| hx₁, x2i |> c(M1, M2)−ε.

Asεis arbitrary, we obtain s=c(M1, M2).

Definition 3.2. LetN ≥2. LetM1,· · ·, MN beN closed subspaces ofHwith intersection M =M₁∩ · · · ∩M_N. TheDixmier number associated to (M₁,· · · , M_N) is defined as

c₀(M₁,· · · , M_N) = sup 2

N−1 P

j<kRehm_j, m_ki km₁k²+· · ·+km_Nk² : mj ∈Mj,km₁k²+· · ·+km_Nk² 6= 0 . The Friedrichs number c(M1,· · ·, M_N) associated to (M1,· · ·, M_N) is defined as

c(M₁,· · ·, M_N) = sup 2

N −1 P

j<kRehm_j, mki km₁k²+· · ·+km_Nk² :

mj ∈Mj∩M^⊥,km₁k²+· · ·+km_Nk² 6= 0o

= sup ( 1

N −1 P

j6=khm_j, m_ki PN

j=1hm_j, m_ji :

mj ∈Mj∩M^⊥,km₁k²+· · ·+km_Nk² 6= 0o .

3B. Other parameters and properties of the Friedrichs number. We found con- venient to introduce the following parameters, called the (reduced or not) configuration constants, although they can be expressed in terms of the Dixmier and Friedrichs numbers (see Proposition 3.6, (f)).

Definition 3.3. LetN ≥2. LetM₁,· · ·, M_N beN closed subspaces ofHwith intersection M =M1∩ · · · ∩MN. The number

κ0(M1,· · · , MN) = sup (1

N kPN

j=1m_jk² PN

j=1km_jk² :mj ∈Mj,km₁k²+· · ·+km_Nk² 6= 0 )

is called thenon-reduced configuration constant of (M1,· · ·, MN). The number κ(M₁,· · ·, M_N) = sup

(1 N

kPN j=1mjk² PN

j=1km_jk² :m_j ∈M_j∩M^⊥,km₁k²+· · ·+km_Nk² 6= 0 )

is called theconfiguration constant of (M1,· · ·, MN).

The configuration constant is related to the maximal possible norms of Gramian matri- ces. Recall that theGramian matrix of an N-tuple of vectors (v₁,· · ·, v_N) is the N×N matrix

G(v₁,· · · , v_N) = [hv_i, v_ji]_1≤i,j≤N.

Proposition 3.4. Let N ≥2. Let M₁,· · · , M_N be N closed subspaces of H with inter- section M =M1∩ · · · ∩MN. Then

κ(M₁,· · ·, M_N) = sup 1

N kG(v₁,· · · , v_N)k:v_j ∈M_j∩M^⊥,kv_jk= 1, j= 1,· · · , N

.

Proof. Letm_j ∈M_j∩M^⊥,j = 1,· · ·, N, withkm₁k²+· · ·+km_Nk² 6= 0. Setv_j = _km^mj

jk

if km_jk 6= 0, or v_j = 0 if m_j = 0. Denote x = (km₁k,· · ·,km_Nk) ∈ C^N \ {0}. We have hx,xi=x^tx6= 0 and

1 N

kPN

j=1m_jk² PN

j=1km_jk² = 1 N

DPN

i km_ikv_i,PN

j km_jkv_jE PN

j=1km_jk²

= 1

N

x^tG(v1,· · · , vN)x x^tx .

The conclusion follows by taking the supremum and noting that the Gramian matrix

G(v₁,· · ·, v_N) is a Hermitian matrix.

Consider the product Hilbert spaceH^N which is the Hilbertian direct sum ofN copies of H, with scalar product

hx,yi=h(x₁,· · ·, xN),(y1,· · · , yN)i:=

N

X

j=1

hx_j, yji.

We denote byCthe Cartesian productC=M1× · · · ×MN ⊂H^N, and byDthe diagonal subsetD= diag (H) ={(y, . . . , y) :y∈H} ⊂H^N. Recall thatM =M₁∩ · · · ∩M_N. Lemma 3.5. The projections onto C, D and C∩D are given by

PC(x1,· · ·, xN) = (P1x1,· · · , PNxN),

PD(x1,· · ·, xN) = ((x1+· · ·+xN)/N,· · ·,(x1+· · ·+xN)/N), and, respectively, by

PC∩D(x1,· · ·, x_N) = ((P_Mx1+· · ·+P_Mx_N)/N,· · · ,(P_Mx1+· · ·+P_Mx_N)/N), where (x₁,· · · , x_N)∈H^N.

Proof. The formulae forPC and PD were proved in [29]. For the third one, we note first that

C∩D= diag (M) ={(m,· · · , m) :m∈M}.

Therefore

kx−PC∩Dxk²= dist(x, diag (M))² = inf{

N

X

j=1

kx_j−mk² :m∈M}.

We obtain

kx−PC∩Dxk² =

N

X

j=1

kx_j−PMxjk²+ inf







N

X

j=1

kP_Mxj −mk²:m∈M





 .

The infimum is realized when the gradient is zero, PN

j=1(m−P_Mx_j) = 0, that is when m=N⁻¹PN

j=1PMxj.

Proposition 3.6. Let N ≥2. Let M₁,· · · , M_N be N closed subspaces of H with inter- section M =M1∩ · · · ∩MN. Then

(a) c0(M1,· · · , MN) = 1 if M 6= {0}, while c0(M1,· · ·, MN) = 0 if and only if the subspaces (M₁,· · · , M_N) are pairwise orthogonal;

(b) c(M₁,· · · , M_N) = c(M₁∩M^⊥,· · · , M_N ∩M^⊥) = c₀(M₁∩M^⊥,· · ·, M_N ∩M^⊥), and thusc(M1,· · · , MN) =c0(M1,· · ·, MN) if M ={0};

(c) 0≤c0(M1,· · · , M_N)≤1 and 0≤c(M1,· · ·, M_N)≤1;

(d) _N¹ ≤κ0(M1,· · · , MN)≤1 and _N¹ ≤κ(M1,· · · , MN)≤1;

(e) κ₀(M₁,· · ·, M_N) =c₀(C,D)² and κ(M₁,· · ·, M_N) =c(C,D)²;

(f) c(M1,· · · , MN) = _N−1^N κ(M1,· · · , MN)− _N−1¹ = _N−1^N c(C,D)²− _N−1¹ and similar statements hold forc₀(M₁,· · · , M_N).

Proof. We start by giving the proof of part (e). We have c(C,D)² = sup

| h(m₁, . . . , mN),(y, . . . , y)i_HN|² N(km₁k²+· · ·+km_Nk²)kyk² :

y∈H, y6= 0, m_j ∈M_j∩M^⊥,km₁k²+· · ·+km_Nk²6= 0o

= sup







DPN

j=1m_j, yE2

N(km₁k²+· · ·+km_Nk²)kyk² :

y∈H, y6= 0, m_j ∈Mj∩M^⊥,km₁k²+· · ·+km_Nk²6= 0o

= sup

( kPN j=1m_jk²

N(km₁k²+· · ·+km_Nk²) :

y∈H, y6= 0, m_j ∈Mj∩M^⊥,km₁k²+· · ·+km_Nk²6= 0o

= κ(M1, . . . , MN).

The proof of the equality κ₀(M₁,· · ·, M_N) =c₀(C,D)² is similar.

We prove now that c(M₁,· · ·, M_N) = _N^N₋₁κ(M₁,· · ·, M_N)− _N−1¹ . Indeed, we have c(M₁,· · · , M_N) = sup

2 N −1

P

j<kRehm_j, mki km₁k²+· · ·+km_Nk² :

mj ∈Mj∩M^⊥,km₁k²+· · ·+km_Nk²6= 0o

= sup ( 1

N −1 kP

jm_jk²−P

jkm_jk² km₁k²+· · ·+km_Nk² : m_j ∈M_j∩M^⊥,km₁k²+· · ·+km_Nk²6= 0o

= N

N−1κ(M₁,· · ·, M_N)− 1 N−1. The proof of the equality for c0(M1,· · · , MN) is similar.

The upper boundκ₀(M₁,· · ·, M_N)≤1 in (d) follows from the Cauchy-Schwarz inequal- ity:

km₁+· · ·+mNk² ≤ (km₁k+· · ·+km_Nk)²

≤ N(km₁k²+· · ·+km_Nk²).

For the lower bound κ0(M1,· · ·, MN)≥1/N, notice that we have, form1∈M1\ {0}, κ0(M1,· · · , MN)≥ 1

N

kkm₁k² km₁k² = 1

N.

The inequalities for κ(M₁,· · ·, M_N) follow from

κ(M1,· · ·, MN) =κ0(M1∩M^⊥,· · · , MN ∩M^⊥).

Now (c) is a consequence of (f) and (d), while (b) and (a) are easy to prove. For the first equality in (b) notice that∩^N_j=1(Mj∩M^⊥) ={0}.

Proposition 3.7. Let N ≥2. Let M₁,· · · , M_N be N closed subspaces of H with inter- section M =M1∩ · · · ∩MN. Then

κ(M₁,· · · , M_N) =

P₁+· · ·+P_N

N −P_M

and

c(M₁,· · ·, M_N) = N N −1

P₁+· · ·+P_N

N −P_M

− 1

N−1. Proof. We have (see for instance [17])

c(C,D) =kP_DPC−PC∩Dk. Using Lemma 3.5,P_DP_C−PC∩D can be written as

1 N











1 1 · · · 1 1 1 · · · 1 ... ... ... 1 1 · · · 1





















P₁ 0 · · · 0 . ..

. ..

0 0 · · · P_N











− 1 N











PM PM · · · PM

P_M · · · P_M

... ...

PM PM · · · PM









 .

Therefore

P_DP_C−PC∩D= 1 N











P1−PM P2−PM · · · PN −PM

P₁−P_M P₂−P_M · · · P_N −P_M

... ...

P1−P_M P2−P_M · · · P_N −P_M









 ,

and so

κ:=κ(M1, . . . , MN) =c(C,D)² = 1 N²











P1−PM P2−PM · · · PN −PM

P₁−P_M P₂−P_M · · · P_N −P_M

... ...

P₁−P_M P₂−P_M · · · P_N −P_M











2

.

We obtain that N²κis equal to











P1−PM P2−PM · · · PN−PM

P₁−P_M P₂−P_M · · · P_N−P_M

... ...

P₁−P_M P₂−P_M · · · P_N−P_M





















P1−PM P1−PM · · · P1−PM

P₂−P_M P₂−P_M · · · P₂−P_M

... ...

P_N−P_M P_N −P_M · · · P_N −P_M









 .

Therefore

N²κ=











Σ Σ · · · Σ Σ Σ · · · Σ ... ... Σ Σ · · · Σ











, where Σ :=

N

X

j=1

(Pj −PM)².

Since

(Pj−PM)² =Pj−PjPM −PMPj +PM = (I−PM)Pj, we have

Σ = (I−P_M)

N

X

j=1

P_j.

LetK be the matrix having all entries equal to Σ. One way to compute the norm of K is to note that, like every circulant matrix, K is unitarily equivalent to a diagonal matrix.

Indeed, denote byF theN×N unitary matrix representing the discrete Fourier transform F = N^−1/2[(ω^jk)]0≤j,k≤N−1, where ω = exp(−2iπ/N) is a primitive Nth root of unity.

Then

F^∗KF =











NΣ 0 · · · 0 0 0 · · · 0 ... ... 0 0 · · · 0









 .

Therefore

κ= 1

NkΣk=k(I−PM)

N

X

j=1

Pjk=k

N

X

j=1

Pj(I−PM)k.

Since PjPM =PM, this can be written as κ=

PN j=1P_j

N −PM

.

The proof is complete.

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. LetN ≥2. LetM₁,· · ·, M_N beN closed subspaces ofHwith intersection M =M1∩ · · · ∩MN. The number

`(M₁, . . . , M_N) = inf

x /∈M

max1≤j≤Ndist(x, Mj) dist(x, M) is called theinclination of (M₁,· · · , M_N).

Proposition 3.9. Let N ≥2. Let M₁,· · · , M_N be N closed subspaces of H with inter- section M =M1∩ · · · ∩MN. Then

1−`(M<sub>1</sub>, . . . , M<sub>N</sub>)≤c(C,D) =κ(M<sub>1</sub>, . . . , M<sub>N</sub>)<sup>1/2</sup>≤1−`(M₁, . . . , M_N)² 2N and

1− 2N

N−1`(M<sub>1</sub>, . . . , M<sub>N</sub>)≤c(M<sub>1</sub>, . . . , M<sub>N</sub>)≤1−`(M₁, . . . , M_N)² N −1 .

In particular, `(M1, . . . , M_N) = 0 if and only if c(M1, . . . , M_N) = 1, if and only if κ(M₁, . . . , M_N) = 1.

Proof. Denote ` = `(M₁, . . . , M_N). Let ε > 0. There exists x ∈ H with kx−P_Mxk = dist(x, M) = 1 such that dist(x, Mj)< `+εfor eachj. Setuj =Pj(x−P<sub>M</sub>x) =Pjx−P<sub>M</sub>x, where Pj is the orthogonal projection ontoMj (1≤ j ≤N). Thenuj ∈Mj and ku<sub>j</sub>k ≤ kx−P<sub>M</sub>xk= 1. Recall thatCis the`²-direct sumC=M₁× · · · ×M_N ⊂H^N, whileD is the diagonal D = diag (H) ={(y, . . . , y) :y ∈H} ⊂H^N. We have C∩D= diag (M), and so y = (y1,· · · , yN) ∈ diag (M)^⊥ if and only if hy₁+· · ·+yN, mi = 0 for every m∈M. Thus (y1,· · · , yN)∈ diag (M)^⊥ if and only ify1+· · ·+yN ∈M^⊥.

Consider

d= ( 1

√N(I−P_M)x, . . . , 1

√N(I −P_M)x)∈D∩ diag (M)^⊥ and

c= ( 1

√Nu₁, . . . , 1

√Nu_N)∈C.

For eachm∈M we have

* _N X

j=1

uj, m +

= 1

√N





N

X

j=1

hP_jx, mi −NhP_Mx, mi





= 1

√ N





N

X

j=1

hx, P_jmi −Nhx, mi





= 0.

Thereforec∈C∩ diag (M)^⊥. We also havekdk= 1 andkck² = _N¹(ku₁k²+· · ·+ku_Nk²)≤ 1. Thus

c(C,D)≥ |hc,di|= 1 N

* _N X

j=1

u_j, x−P_Mx +

≥ 1 N Re

* _N X

j=1

u_j, x−P_Mx +

.

For a fixed j we have kx−P_Mx−ujk=kx−Pjxk= dist(x, Mj)< `+ε. Therefore 2 Rehx−P<sub>M</sub>x, u<sub>j</sub>i=kx−P<sub>M</sub>xk<sup>2</sup>+ku<sub>j</sub>k<sup>2</sup>− kx−P<sub>M</sub>x−u<sub>j</sub>k<sup>2</sup> >1 +ku<sub>j</sub>k<sup>2</sup>−(`+ε)². We also have ku_jk ≥ kx−PMxk − kx−PMx−ujk>1−(`+ε). We obtain

c(C,D) ≥ 1 N Re

* _N X

j=1

u_j, x−P_Mx +

≥ 1

2N

N

X

j=1

1 +ku_jk²−(`+ε)²

≥ 1

2N N +N(1−(`+ε))<sup>2</sup>−N(`+ε)²

= 1

2 1 + (1−(`+ε))<sup>2</sup>−(`+ε)² . As this inequality is true for everyε >0 we get

c(C,D)≥ 1−`<sup>2</sup>+ (1−`)²

2 = 1−`.

Denotec=c(C,D). Letε >0. There existc∈C∩ diag (M)^⊥andd∈D∩ diag (M)^⊥ with kck = 1, kdk = 1 such that c < | hc,di |+ε. Let θ ∈ R be such that hc,di = e^iθ| hc,di |. Then

kc−e^iθdk² = 2−2 Re(e^−iθhc,di) = 2−2| hc,di | ≤2−2(c−ε).

Set c= (m₁, . . . , m_N) ∈C∩ diag (M)^⊥ and e^iθd = (y, . . . , y) ∈D∩ diag (M)^⊥. Then y∈M^⊥ and

km₁k²+· · ·+km_Nk²= 1, kyk= 1/

√

N , and

N

X

j=1

km_j−yk²≤2−2(c−ε).

Letx=√

N y. Thenx∈M^⊥, dist(x, M) =kxk= 1 and we have dist(x, M_j)²≤ kx−√

N m_jk²=Nkm_j−yk²≤N(2−2(c−ε)).

We finally obtain `² ≤2N(1−c), and so

1−`≤c(C,D)≤1− `² 2N. Using the equalities

c(C,D) =κ(M1, . . . , MN)^1/2 and κ(M1, . . . , MN) = N−1

N c(M1,· · · , MN) + 1 N we obtain

N(1−`)²−1

N −1 ≤c(M1,· · ·, MN)≤ N(1−(`²/2N)²)−1

N−1 .

Since 2−`≤2, we can write N(1−`)²−1

N−1 = 1− N

N−1`(2−`)≥1− 2N N −1`.

We also have

N(1−(`²/2N)²)−1

N−1 = 1− `²

N −1(1− `²

4N)≤1− `² N−1.

4. Characterising (ASC) for products of projections

4A. A qualitative result. When T is the product of N orthogonal projections, we know from Theorem 2.4 that the dichotomy (QUC)/(ASC) holds, and that we have quick uniform convergence if and only if the range of T−I is closed. The following qualitative result gives a characterization of the (ASC) condition in terms of several parameters associated to (M₁,· · · , M_N), or spectral properties ofT, or random products. We denote by k · k_e theessential norm and by σ_e theessential spectrum.

Theorem 4.1. Let N ≥2. LetM1,· · ·, MN beN closed subspaces of H with intersection M =M₁∩ · · · ∩M_N. Denote P_j the orthogonal projection onto M_j, 1 ≤j ≤ N, and by P_M the orthogonal projection onto M. Let T =P_NPN−1· · ·P₁. The following assertions are equivalent:

(1) Ran(T −I) is not closed;

(1⁰) for every k ≥ N and every sequence of indices (i_k)k≥1 such that {i₁, . . . , i_k} = {1,2, . . . , N}, Ran(Pik· · ·Pi1 −I) is not closed;

(2) one of the conditions (ASC1), (ASC2), (ASC3), (ASCH) holds for T;

(2⁰) (ASCH) for random products: for everyε >0, every sequence (a_n)n≥0 of positive reals with limn→∞an= 0, and every sequence of indices (ik)k≥1 in {1,2, . . . , N}, there existsx∈H with kxk<sup_nan+ε such that

Re

P_inP_in−1· · ·P_i1x−P_Mx, x

> a_n for eachn≥1;

(3) c(M1,· · · , M_N) = 1. Equivalently, κ(M1,· · ·, M_N) = 1, or `(M1,· · · , M_N) = 0;

(4) for every ε >0, every closed subspace K ⊂ M^⊥ of finite codimension (in M^⊥), there existsx∈K such that kxk= 1 andmax{dist(x, M_j) :j= 1,· · ·, N}< ε;

(5) 1∈σ(T−PM);

(5⁰) for everyk and every i₁,· · ·, i_k ∈ {1,2,· · · , N} we have 1∈σ(P_ik· · ·P_i1 −P_M);

(6) kT−P_Mk= 1 ;

(6⁰) for everyk and every sequence of indices (i_k)k≥1, 1≤i_k≤N, with {i₁, . . . , i_k}= {1,2, . . . , N} we have kP_ik· · ·P_i1−P_Mk= 1 ;

(7) kT−PMk_e = 1;

(7⁰) for everyk and every i₁,· · ·, i_k ∈ {1,2,· · · , N} we have kP_ik· · ·P_i1−P_Mk_e= 1;

(8) 1∈σe(T −PM);

(8⁰) for everyk, every i₁,· · · , i_k∈ {1,2,· · · , N} we have 1∈σ_e(P_ik· · ·P_i1 −P_M);

(9) for every ε >0, every closed subspace K ⊂ M^⊥ of finite codimension (in M^⊥), there existsx∈K such that kT x−xk ≤ε;

(9⁰) for every ε >0, every closed subspace K ⊂ M^⊥ of finite codimension (in M^⊥), there exists x∈K such that kP_ik· · ·P_i1x−xk ≤εfor every k, every i₁,· · ·, i_k ∈ {1,2,· · ·, N} ;

(10) the sum of diag(M₁)⊂H^N−1 and M₂⊕ · · · ⊕M_N ⊂H^N−1 is not closed in H^N−1 (and equivalent statements for diag(Mj)⊂H^N−1, 2≤j≤N);

(11) M₁^⊥+· · ·+M_N^⊥ is not closed in H.

The conditions (1),(2),(5),(6),(7),(8) and (9), most of them of spectral nature, are conditions about T =P_N· · ·P1, while the corresponding conditions denoted with primes are analog conditions about random productsP_iN· · ·P_i1. The conditions (3),(4),(10) and (11) are about the geometry of subspaces Mj.

Notice that we have the dichotomy (QUC)/(ASC) in all possible senses, and that (QUC) holds if and only if c(M₁, . . . , M_N)<1. A quantitative estimate reflecting the geometric convergence ofkTⁿ−P_Mk to zero, in terms of the Friedrichs number, will be given after the proof of the theorem.

Proof of Theorem 4.1. ”(1)⇔(2)“ The equivalence of (1) and (2) follows from Theorem 2.4.

”(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^⊥, we have 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{kx−T xk:x∈H,dist(x,Ker(T −I) = 1}

thereduced minimum modulus of T −I. Then Ran(T −I) is closed if and only if γ >0.

Clearly T y=y fory∈M. IfT x=x, then

kxk=kP_N· · ·P1xk ≤ kP_N−1· · ·P1xk ≤ · · · ≤ kP₁xk ≤ kxk.

We successively obtain P₁x = x, P₂x = x, . . . , P_Nx = x, and finally x ∈ M. Thus Ker(T−I) =M.

Letε >0. There existsx∈H withkx−P_Mxk= dist(x, M) = 1 such thatkx−T xk ≤ γ+ε. We obtain

1 =kx−P_Mxk ≥ kP₁(x−P_Mx)k=kP₁x−P_Mxk ≥ kP₂P₁x−P_Mxk

≥ · · · ≥ kP_N· · ·P₁x−P_Mxk=kT x−P_Mxk

≥ kx−PMxk − kx−T xk ≥1−γ−ε.

We also have

k(I−P₁)(x−P_Mx)k² = kx−P_Mxk²− kP₁x−P_Mxk²

≤ 1−(1−γ−ε)² =−(γ+ε)²+ 2(γ+ε)≤2γ+ 2ε.

Thus dist(x, M₁) =kx−P₁xk=k(I−P₁)(x−P_Mx)k ≤(2γ+ 2ε)^1/2.

Lety=x−PMx; thenkyk= 1. For a fixed sbetween 1 andN we can write kP_s· · ·P1y−Ps+1· · ·P1yk² = kP_s· · ·P1yk²− kP_s+1· · ·P1yk²

≤ kyk²− kP_s+1· · ·P1x−PMxk²

≤ 1−(1−γ−ε)²≤2γ+ 2ε.

Thus

dist(x, Mj) = dist(y, Mj)≤ ky−Pj· · ·P1yk

≤ ky−P₁yk+kP₁y−P₂P₁yk+· · ·+kP_j−1· · ·P₁y−P_j· · ·P₁yk

≤ jp

(2γ+ 2ε),

for everyj. Hence max1≤j≤Ndist(x, M_j)≤Np

(2γ+ 2ε) and, asεis arbitrary,

`:=`(M₁, . . . , M_N)≤Np 2γ.

We obtain _2N¹2`(M<sub>1</sub>, . . . , M<sub>N</sub>)<sup>2</sup> ≤ γ(T −I). Therefore Ran(I −T) not closed (γ = 0) implies `(M₁, . . . , M_N) = 0.

”(3) ⇒ (1⁰)“ Letk ≥N and let (ik)k≥1 be a sequence of indices with {i₁, . . . , ik}= {1,2, . . . , N}. This implies that Ker(I −Pi_kPik−1· · ·Pi1) = M. Let ` = `(M1, . . . , M_N) and let ε > 0. There exists x ∈ H with kx −PMxk = dist(x, M) = 1 such that maxjdist(x, Mj)< `+ε. We have

kx−P_i1xk= dist(x, M_i1)< `+ε and

kP_i2Pi1x−Pi1xk= dist(Pi1x, Mi2)≤ kx−Pi1xk+ dist(x, Mi2)<2(`+ε).

Setx0=x and xs =PisPis−1· · ·Pi1x fors≥1. Suppose that (4.1) kx_s−xs−1k ≤2^s−1(`+ε) holds for 1≤s≤r. Then

kx_r+1−x_rk = dist(P_ir· · ·P_i1x, M_ir+1)

≤ kP_ir· · ·Pi1x−xk+ dist(x, Mir+1)

≤ kx_s−xs−1k+kx_s−1−xs−2k+· · ·+kx₁−xk + dist(x, M_r+1)

≤ (2^r−1+ 2^r−2+· · ·+ 2 + 1 + 1)(`+ε) = 2<sup>r</sup>(`+ε).