The rate of convergence in the method of alternating projections

Introduction

Throughout the paper $H$ is a complex Hilbert space. For a closed linear subspace $S$ of $H$ we denote by $S^{\perp}$ its orthogonal complement in $H$, and by $P_{S}$ the orthogonal projection of $H$ onto $S$. In this paper $N$ denotes a fixed positive integer greater or equal than $2$.

It was proved by J. von Neumann [27, p. 475] that for two closed subspaces $M_{1}$ and $M_{2}$ of $H$, with intersection $M=M_{1}\cap M_{2}$, the following convergence result holds:

Using the notation $T=P_{M_{2}}P_{M_{1}}$, von Neumann’s result says that the iterates $T^{n}$ of $T$ are strongly convergent to $T^{\infty}=P_{M}$. 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 $M_{1},\dots,M_{N}$ with intersection $M=M_{1}\cap M_{2}\cdots\cap M_{N}$ was proved by Halperin : for each $x\in H$ we have

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 : if $X$ is a uniformly convex Banach space and $P_{j}$, $1\leq j\leq N$, are $N$ norm one projections in ${\mathcal{B}}(X)$, then the iterates of $T=P_{N}\cdots P_{2}P_{1}$ are strongly convergent. The strong limit $T^{\infty}$ is a projection of norm one onto the intersection of the ranges of $P_{j}$. The same result holds if $X$ is uniformly smooth and each projection $P_{j}$ is of norm one. It also holds if $X$ is a reflexive (complex) Banach space and each projection $P_{j}$ is hermitian (that is, with real numerical range). We refer to and the references therein for other Banach space results of this type.

An interesting extension of the method of cyclic alternating projections is the method of random alternating projections. Let $P_{j}$, $1\leq j\leq N$, be $N$ orthogonal projections in ${\mathcal{B}}(H)$, $M=\cap_{j=1}^{N}\operatorname{Ran}(P_{j})$, and let $(i_{k})_{k\geq 1}$ be a sequence from $\{1,2,\dots,N\}$ (random samples). The method of random alternating projections asks about the convergence of the sequence $(x_{n})_{n\geq 0}$ given by $x_{0}=x$, $x_{n}=P_{i_{n}}x_{n-1}$. It is an open problem to know whether $(x_{n})_{n\geq 0}$ is always convergent in the topology of $H$. The convergence of $(x_{n})_{n\geq 0}$ in the weak topology has been proved by Amemiya and Ando . If each $j$ between $1$ and $N$ occurs infinitely many times in the sequence of random samples, then the weak limit of $(x_{n})_{n\geq 0}$ is $P_{M}x$. We refer to for results related to this problem.

B. 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.

(Friedrichs angle) Let $M_{1}$ and $M_{2}$ be two closed subspaces of the Hilbert space $H$ with intersection $M=M_{1}\cap M_{2}$. The Friedrichs angle between the subspaces $M_{1}$ and $M_{2}$ is defined to be the angle in $[0,\pi/2]$ whose cosine is given by

where $B_{H}:=\{h\in H:\|h\|\leq 1\}$ is the unit ball of $H$. The minimal angle (or Dixmier angle) between the subspaces $M_{1}$ and $M_{2}$ is defined to be the angle in $[0,\pi/2]$ whose cosine is given by

We note that $c(M_{1},M_{2})=c_{0}(M_{1}\cap M^{\perp},M_{2}\cap M^{\perp})$, and that $c_{0}(M_{1},M_{2})=1$ if $M\neq\{0\}$. We also have $c(M_{1},M_{2})=c(M_{1}^{\perp},M_{2}^{\perp})$. We refer to the survey paper 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 (upper bound) and by Kayalar and Weinert (equality) that

This formula shows that the sequence $(T^{n})$ of iterates of $T=P_{M_{2}}P_{M_{1}}$ converges uniformly to $T^{\infty}=P_{M}$ if and only if $c(M_{1},M_{2})<1$, i.e., if the Friedrichs angle between $M_{1}$ and $M_{2}$ is positive. When this happens, the iterates of $T=P_{M_{2}}P_{M_{1}}$ converge “quickly” (i.e. at the rate of a geometrical progression) to $T^{\infty}=P_{M}$, in the following sense:

(quick uniform convergence) there exist $C>0$ and $\alpha\in]0,1[$ such that

It is also known that $c(M_{1},M_{2})<1$ if and only if $M_{1}+M_{2}$ is closed, if and only if $M_{1}^{\perp}+M_{2}^{\perp}$ is closed, if and only if $(M_{1}\cap M^{\perp})+(M_{2}\cap M^{\perp})$ is closed.

When $M_{1}+M_{2}$ is not closed, we have strong, but not uniform convergence. It was recently proved by Bauschke, Deutsch and Hundal (see 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 for $N\geq 3$ are not as complete as the results described above for $N=2$. We refer to , [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 $N\geq 3$ 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.

C. 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 ($N\geq 3$). 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 dichotomy 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 dichotomy (QUC)/(ASC) in all above-mentioned generalizations of the method of alternating projections. We also give a generalization of the Friedrichs angle to several subspaces, $c(M_{1},\cdots,M_{N})$, and prove that condition (QUC) holds in Halperin’s theorem if and only if $c(M_{1},\cdots,M_{N})<1$. Estimates for the error $\|(P_{M_{N}}\cdots P_{M_{2}}P_{M_{1}})^{n}-P_{M}\|$ are given in this case and several statements equivalent to the condition $c(M_{1},\cdots,M_{N})<1$ are obtained. Some of them are expressed in terms of random products $P_{i_{k}}\cdots P_{i_{1}}$ of projections. More specific descriptions of these results, and information about how the paper is organized, are given below.

D. Conditions for arbitrarily slow convergence

Several dichotomy theorems of the type quick uniform convergence versus 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):

(arbitrarily slow convergence, variant 1) for every $\varepsilon>0$ and every sequence $(a_{n})_{n\geq 1}$ of positive numbers such that $\lim_{n\to\infty}a_{n}=0$, there exists a vector $x\in X$ such that $\|x\|<\sup_{n}a_{n}+\varepsilon$ and $\|T^{n}x-T^{\infty}x\|\geq a_{n}$ for all $n$.

(arbitrarily slow convergence, variant 2) for every sequence $(a_{n})_{n\geq 1}$ of positive numbers such that $\lim_{n\to\infty}a_{n}=0$, there exists a dense subset of points $x\in X$ such that $\|T^{n}x-T^{\infty}x\|\geq a_{n}$ for all but a finite number of $n$’s.

(arbitrarily slow convergence, variant 3) for every sequence $(a_{n})_{n\geq 1}$ of positive numbers such that $\lim_{n\to\infty}a_{n}=0$, there exist two vectors $x\in X$ and $y\in X^{*}$ (the dual of $X$) such that $\operatorname{Re}\left\langle T^{n}x-T^{\infty}x,y\right\rangle\geq a_{n}$ for all $n\geq 1$. Furthermore, if there is a Banach space $Y$ such that $X$ is a (isometrical) subspace of $Y^{*}$, then the vector $y$ can be chosen in $Y$;

(arbitrarily slow convergence, Hilbertian version) for every $\varepsilon>0$ and every sequence $(a_{n})_{n\geq 1}$ of positive numbers such that $\lim_{n\to\infty}a_{n}=0$, there exists a vector $x\in H$ such that $\|x\|<\sup_{n}a_{n}+\varepsilon$ and $\operatorname{Re}\left\langle T^{n}x-T^{\infty}x,x\right\rangle\geq a_{n}$ for all $n\geq 1$ . Here $H$ 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 ).

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

Let $X$ be a Banach space which does not contain $c_{0}$, and $T$ a bounded operator on $X$ such that $1$ belongs to the spectrum of $T$ and $\|T^{n}x\|$ tends to zero as $n$ tends to infinity for every $x\in X$. Then for any sequence $(a_{n})_{n\geq 0}$ such that $a_{n}$ tends to zero as $n$ tends to infinity, there exists a vector $x\in X$ and a functional $x^{*}\in X^{*}$ such that $\operatorname{Re}\langle T^{n}x,x^{*}\rangle\geq a_{n}$ for every $n\geq 0$.

E. A generalization of the Friedrichs angle

In order to quantify the rate of convergence in the method of alternating projections, an extension of the cosine of Friedrichs angle to several subspaces $(M_{1},\dots,M_{N})$ will be given in Section 3. It is a parameter $c(M_{1},\dots,M_{N})$ which lies between $0$ and $1$, defined as follows:

In Section $4$ we characterize in several ways when the dichotomy (QUC)/(ASC) arises. The characterizations are in terms of geometric properties of $(M_{1},\cdots,M_{N})$, of spectral properties of $T$, or of random products $P_{i_{k}}\cdots P_{i_{1}}$. We give an estimate for the geometric convergence of $\|T^{n}-P_{M}\|$ to zero when $c(M_{1},\dots,M_{N})<1$.

General dichotomy theorems and applications

Let $X$ be a Banach space and let $T\in{\mathcal{B}}(X)$ be such that the sequence of iterates $(T^{n})$ is strongly convergent to $T^{\infty}\in{\mathcal{B}}(X)$. Then the following dichotomy holds : either (QUC), or (ASC1). The quick uniform convergence (condition (QUC)) holds if and only if

In these statements, the condition (ASC1) can be replaced by (ASC2).

Suppose that the sequence of iterates $(T^{n})_{n\geq 0}$ is strongly convergent to $T^{\infty}\in{\mathcal{B}}(X)$. Then $T$ is mean ergodic, i.e., the Cesàro means $(I+T+\cdots+T^{n-1})/n$ 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, $X=\operatorname{Ker}(T-I)\oplus\overline{\operatorname{Ran}(T-I)}$. Moreover, $T^{\infty}$ is the projection onto $\operatorname{Ker}(T-I)$ along $\overline{\operatorname{Ran}(T-I)}$. Notice also that $T^{\infty}$ acts on the space $\operatorname{Ker}(T-I)$ as the identity. With respect to the decomposition $X=\operatorname{Ker}(T-I)\oplus\overline{\operatorname{Ran}(T-I)}$ we can write

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

Since $r(A)<1$, there exist $C>0$ and $\alpha\in]0,1[$ such that

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

Case (2). We have $r(A)=1$. Recall that $\|A^{n}y\|\to 0$ as $n\to\infty$, for each $y\in\overline{\operatorname{Ran}(T-I)}$. The conditions (ASC1) and (ASC2) follow now from [26, Thm 14, p. 333].

The following is a different argument for the last part of the proof, without the use of Fredholm theory. As $\lambda\in\sigma(A)\setminus\sigma_{p}(A)$ and $\operatorname{Ran}(A-\lambda)$ is closed, the operator $A-\lambda$ is lower bounded, and thus $\lambda$ 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 as a basic reference for the spectral theory of linear operators we are using in the present paper.

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

B. Applications to the method of alternating projections

We introduce first some notation, and recall for the convenience of the reader some Banach space terminology. Let $N\geq 2$. Let $X$ be a Banach space and let $P_{1},\cdots,P_{N}$ be $N$ fixed projections ($P_{j}^{2}=P_{j}$) acting on $X$. We denote by $\mathcal{S}=\mathcal{S}(P_{1},\cdots,P_{N})$ the convex multiplicative semigroup generated by $P_{1},\cdots,P_{N}$. Recall that this is the convex hull of the set of all products with factors from $P_{1},\cdots,P_{N}$, and that the convex hull of every multiplicative semigroup of operators is a semigroup.

The space $X$ is said to be uniformly convex if for every $\varepsilon\in(0,1)$ there exists $\delta\in(0,1)$ such that for any two vectors, $x$ and $y$, with $\|x\|\leq 1$ and $\|y\|\leq 1$, $\|x+y\|/2>1-\delta$ implies $\|x-y\|<\varepsilon$. An (equivalent) definition of a uniformly smooth Banach space is the following: $X$ is uniformly smooth if its dual, $X^{*}$, is uniformly convex. We refer to for more information.

We call $P\in{\mathcal{B}}(X)$ a norm one projection (non-zero orthoprojection) if $P^{2}=P$ and $\|P\|=1$. 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 $\|\exp(itT)\|=1$ for every real $t$. Hermitian operators on Hilbert spaces coincide with the self-adjoint ones; see for instance and the references therein.

Let $N\geq 2$. Let $X$ be a complex Banach space, and let $P_{1},\cdots,P_{N}$ be $N$ projections on $X$. Let $T$ be an operator in $\mathcal{S}(P_{1},\cdots,P_{N})$. If one of the following conditions below 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 a Hilbert space) applies:

the space $X$ is uniformly convex and each $P_{j}$, $1\leq j\leq N$, is a norm one projection;

the space $X$ is uniformly smooth, and each $P_{j}$, $1\leq j\leq N$, is a norm one projection;

the space $X$ is reflexive and for each $j$ there exists $r_{j}$ with $0<r_{j}<1$ such that $\|P_{j}-r_{j}I\|\leq 1-r_{j}$. In particular, this holds if each $P_{j}$ is hermitian, $1\leq j\leq N$.

A generalization of Friedrichs angle for NN subspaces

As mentioned in the introduction, the rate of convergence in the method of alternating projections for two closed subspaces $M_{1}$ and $M_{2}$ is controlled by the Friedrichs angle $c(M_{1},M_{2})$. We introduce and study in this section a generalization of Friedrichs angle for $N$ subspaces.

In order to introduce our generalization of the cosine of the Friedrichs angle to several closed subspaces, we start by giving an equivalent definition of the Friedrichs angle $c(M_{1},M_{2})$.

(a) Let $M_{1}$ and $M_{2}$ be two closed subspaces of $H$. Then

(b) Let $M_{1}$ and $M_{2}$ be two closed subspaces in $H$. Then

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 $(m_{1},m_{2})$ with $(m_{1},m_{2})\neq(0,0)$ we have

As $\varepsilon$ is arbitrary, we obtain $s=c(M_{1},M_{2})$. ∎

Let $N\geq 2$. Let $M_{1},\cdots,M_{N}$ be $N$ closed subspaces of $H$ with intersection $M=M_{1}\cap\cdots\cap M_{N}$. The Dixmier number associated to $(M_{1},\cdots,M_{N})$ is defined as

The Friedrichs number $c(M_{1},\cdots,M_{N})$ associated to $(M_{1},\cdots,M_{N})$ is defined as

B. Other parameters and properties of the Friedrichs number.

We found convenient 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)).

Let $N\geq 2$. Let $M_{1},\cdots,M_{N}$ be $N$ closed subspaces of $H$ with intersection $M=M_{1}\cap\cdots\cap M_{N}$. The number

is called the non-reduced configuration constant of $(M_{1},\cdots,M_{N})$. The number

is called the configuration constant of $(M_{1},\cdots,M_{N})$.

The configuration constant is related to the maximal possible norms of Gramian matrices. Recall that the Gramian matrix of an $N$-tuple of vectors $(v_{1},\cdots,v_{N})$ is the $N\times N$ matrix

Let $N\geq 2$. Let $M_{1},\cdots,M_{N}$ be $N$ closed subspaces of $H$ with intersection $M=M_{1}\cap\cdots\cap M_{N}$. Then

The conclusion follows by taking the supremum and noting that the Gramian matrix $G(v_{1},\cdots,v_{N})$ is a Hermitian matrix. ∎

Consider the product Hilbert space $H^{N}$ which is the Hilbertian direct sum of $N$ copies of $H$, with scalar product

We denote by $\mathbf{C}$ the Cartesian product $\mathbf{C}=M_{1}\times\cdots\times M_{N}\subset H^{N}$, and by $\mathbf{D}$ the diagonal subset $\mathbf{D}=\mbox{ diag }(H)=\{(y,\dots,y):y\in H\}\subset H^{N}$. Recall that $M=M_{1}\cap\cdots\cap M_{N}$.

The projections onto $\mathbf{C}$, $\mathbf{D}$ and $\mathbf{C}\cap\mathbf{D}$ are given by

The formulae for $P_{\mathbf{C}}$ and $P_{\mathbf{D}}$ were proved in . For the third one, we note first that

The infimum is realized when the gradient is zero, $\sum_{j=1}^{N}(m-P_{M}x_{j})=0$, that is when $m=N^{-1}\sum_{j=1}^{N}P_{M}x_{j}$. ∎

Let $N\geq 2$. Let $M_{1},\cdots,M_{N}$ be $N$ closed subspaces of $H$ with intersection $M=M_{1}\cap\cdots\cap M_{N}$. Then

$c_{0}(M_{1},\cdots,M_{N})=1$ if $M\neq\{0\}$, while $c_{0}(M_{1},\cdots,M_{N})=0$ if and only if the subspaces $(M_{1},\cdots,M_{N})$ are pairwise orthogonal;

$c(M_{1},\cdots,M_{N})=c(M_{1}\cap M^{\perp},\cdots,M_{N}\cap M^{\perp})=c_{0}(M_{1}\cap M^{\perp},\cdots,M_{N}\cap M^{\perp})$, and thus $c(M_{1},\cdots,M_{N})=c_{0}(M_{1},\cdots,M_{N})$ if $M=\{0\}$;

$0\leq c_{0}(M_{1},\cdots,M_{N})\leq 1$ and $0\leq c(M_{1},\cdots,M_{N})\leq 1$;

$\frac{1}{N}\leq\kappa_{0}(M_{1},\cdots,M_{N})\leq 1$ and $\frac{1}{N}\leq\kappa(M_{1},\cdots,M_{N})\leq 1$;

$\kappa_{0}(M_{1},\cdots,M_{N})=c_{0}(\mathbf{C},\mathbf{D})^{2}$ and $\kappa(M_{1},\cdots,M_{N})=c(\mathbf{C},\mathbf{D})^{2}$;

$c(M_{1},\cdots,M_{N})=\frac{N}{N-1}\kappa(M_{1},\cdots,M_{N})-\frac{1}{N-1}=\frac{N}{N-1}c(\mathbf{C},\mathbf{D})^{2}-\frac{1}{N-1}$ and similar statements hold for $c_{0}(M_{1},\cdots,M_{N})$.

We start by giving the proof of part (e). We have

The proof of the equality $\kappa_{0}(M_{1},\cdots,M_{N})=c_{0}(\mathbf{C},\mathbf{D})^{2}$ is similar.

We prove now that $c(M_{1},\cdots,M_{N})=\frac{N}{N-1}\kappa(M_{1},\cdots,M_{N})-\frac{1}{N-1}$. Indeed, we have

The proof of the equality for $c_{0}(M_{1},\cdots,M_{N})$ is similar.

The upper bound $\kappa_{0}(M_{1},\cdots,M_{N})\leq 1$ in (d) follows from the Cauchy-Schwarz inequality:

For the lower bound $\kappa_{0}(M_{1},\cdots,M_{N})\geq 1/N$, notice that we have, for $m_{1}\in M_{1}\setminus\{0\}$,

The inequalities for $\kappa(M_{1},\cdots,M_{N})$ follow from

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 $\cap_{j=1}^{N}(M_{j}\cap M^{\perp})=\{0\}$. ∎

Let $N\geq 2$. Let $M_{1},\cdots,M_{N}$ be $N$ closed subspaces of $H$ with intersection $M=M_{1}\cap\cdots\cap M_{N}$. Then

Using Lemma 3.5, $P_{\mathbf{D}}P_{\mathbf{C}}-P_{\mathbf{C}\cap\mathbf{D}}$ can be written as

Let $K$ be the matrix having all entries equal to $\Sigma$. 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 by $F$ the $N\times N$ unitary matrix representing the discrete Fourier transform $F=N^{-1/2}[(\omega^{jk})]_{0\leq j,k\leq N-1}$, where $\omega=\exp(-2i\pi/N)$ is a primitive $N$th root of unity. Then

Since $P_{j}P_{M}=P_{M}$, this can be written as

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 , and the references therein.

Let $N\geq 2$. Let $M_{1},\cdots,M_{N}$ be $N$ closed subspaces of $H$ with intersection $M=M_{1}\cap\cdots\cap M_{N}$. The number

is called the inclination of $(M_{1},\cdots,M_{N})$.

Let $N\geq 2$. Let $M_{1},\cdots,M_{N}$ be $N$ closed subspaces of $H$ with intersection $M=M_{1}\cap\cdots\cap M_{N}$. Then

Therefore $\mathbf{c}\in\mathbf{C}\cap\mbox{ diag }(M)^{\perp}$. We also have $\|\mathbf{d}\|=1$ and $\|\mathbf{c}\|^{2}=\frac{1}{N}(\|u_{1}\|^{2}+\cdots+\|u_{N}\|^{2})\leq 1$. Thus

As this inequality is true for every $\varepsilon>0$ we get

Set $\mathbf{c}=(m_{1},\dots,m_{N})\in\mathbf{C}\cap\mbox{ diag }(M)^{\perp}$ and $e^{i\theta}\mathbf{d}=(y,\dots,y)\in\mathbf{D}\cap\mbox{ diag }(M)^{\perp}$. Then $y\in M^{\perp}$ and

Let $x=\sqrt{N}y$. Then $x\in M^{\perp}$, $\operatorname{dist}(x,M)=\|x\|=1$ and we have

Characterising (ASC) for products of projections

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 $(M_{1},\cdots,M_{N})$, or spectral properties of $T$, or random products. We denote by $\|\cdot\|_{e}$ the essential norm and by $\sigma_{e}$ the essential spectrum.

Let $N\geq 2$. Let $M_{1},\cdots,M_{N}$ be $N$ closed subspaces of $H$ with intersection $M=M_{1}\cap\cdots\cap M_{N}$. Denote $P_{j}$ the orthogonal projection onto $M_{j}$, $1\leq j\leq N$, and by $P_{M}$ the orthogonal projection onto $M$. Let $T=P_{N}P_{N-1}\cdots P_{1}$. The following assertions are equivalent:

for every $k\geq N$ and every sequence of indices $(i_{k})_{k\geq 1}$ such that $\{i_{1},\ldots,i_{k}\}=\{1,2,\ldots,N\}$, $\operatorname{Ran}(P_{i_{k}}\cdots P_{i_{1}}-I)$ is not closed;

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

(ASCH) for random products: for every $\varepsilon>0$, every sequence $(a_{n})_{n\geq 0}$ of positive reals with $\lim_{n\to\infty}a_{n}=0$, and every sequence of indices $(i_{k})_{k\geq 1}$ in $\{1,2,\ldots,N\}$, there exists $x\in H$ with $\|x\|<\sup_{n}a_{n}+\varepsilon$ such that

for every $\varepsilon>0$, every closed subspace $K\subset M^{\perp}$ of finite codimension (in $M^{\perp}$), there exists $x\in K$ such that $\|x\|=1$ and $\max\{\operatorname{dist}(x,M_{j}):j=1,\cdots,N\}<\varepsilon$;

for every $k$ and every $i_{1},\cdots,i_{k}\in\{1,2,\cdots,N\}$ we have $1\in\sigma(P_{i_{k}}\cdots P_{i_{1}}-P_{M})$;

for every $k$ and every sequence of indices $(i_{k})_{k\geq 1}$, $1\leq i_{k}\leq N$, with $\{i_{1},\ldots,i_{k}\}=\{1,2,\ldots,N\}$ we have $\|P_{i_{k}}\cdots P_{i_{1}}-P_{M}\|=1$ ;

for every $k$ and every $i_{1},\cdots,i_{k}\in\{1,2,\cdots,N\}$ we have $\|P_{i_{k}}\cdots P_{i_{1}}-P_{M}\|_{e}=1$;

for every $k$, every $i_{1},\cdots,i_{k}\in\{1,2,\cdots,N\}$ we have $1\in\sigma_{e}(P_{i_{k}}\cdots P_{i_{1}}-P_{M})$;

for every $\varepsilon>0$, every closed subspace $K\subset M^{\perp}$ of finite codimension (in $M^{\perp}$), there exists $x\in K$ such that $\|Tx-x\|\leq\varepsilon$;

for every $\varepsilon>0$, every closed subspace $K\subset M^{\perp}$ of finite codimension (in $M^{\perp}$), there exists $x\in K$ such that $\|P_{i_{k}}\cdots P_{i_{1}}x-x\|\leq\varepsilon$ for every $k$, every $i_{1},\cdots,i_{k}\in\{1,2,\cdots,N\}$ ;

the sum of $\mbox{ diag}(M_{1})\subset H^{N-1}$ and $M_{2}\oplus\cdots\oplus M_{N}\subset H^{N-1}$ is not closed in $H^{N-1}$ (and equivalent statements for $\mbox{ diag}(M_{j})\subset H^{N-1}$, $2\leq j\leq N$);

$M_{1}^{\perp}+\cdots+M_{N}^{\perp}$ is not closed in $H$.

The conditions $(1),(2),(5),(6),(7),(8)$ and $(9)$, most of them of spectral nature, are conditions about $T=P_{N}\cdots P_{1}$, while the corresponding conditions denoted with primes are analog conditions about random products $P_{i_{N}}\cdots P_{i_{1}}$. 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 $c(M_{1},\ldots,M_{N})<1$. A quantitative estimate reflecting the geometric convergence of $\|T^{n}-P_{M}\|$ to zero, in terms of the Friedrichs number, will be given after the proof of the theorem.

”$(1)\Leftrightarrow(2)$“ The equivalence of (1) and (2) follows from Theorem 2.4.

”$(1)\Leftrightarrow(5)$“ 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 $H=M\oplus M^{\perp}$, we have $T=P_{M}\oplus A$, where $A=T\mid_{M^{\perp}}=T(I-P_{M})=T-P_{M}$.

”$(1)\Rightarrow(3)$“ We prove this implication in a quantitative form. Denote

the reduced minimum modulus of $T-I$. Then $\operatorname{Ran}(T-I)$ is closed if and only if $\gamma>0$. Clearly $Ty=y$ for $y\in M$. If $Tx=x$, then

We successively obtain $P_{1}x=x$, $P_{2}x=x$, …, $P_{N}x=x$, and finally $x\in M$. Thus $\operatorname{Ker}(T-I)=M$.

Let $\varepsilon>0$. There exists $x\in H$ with $\|x-P_{M}x\|=\operatorname{dist}(x,M)=1$ such that $\|x-Tx\|\leq\gamma+\varepsilon$. We obtain

Thus $\operatorname{dist}(x,M_{1})=\|x-P_{1}x\|=\|(I-P_{1})(x-P_{M}x)\|\leq(2\gamma+2\varepsilon)^{1/2}$.

Let $y=x-P_{M}x$; then $\|y\|=1$. For a fixed $s$ between $1$ and $N$ we can write

for every $j$. Hence $\max_{1\leq j\leq N}\operatorname{dist}(x,M_{j})\leq N\sqrt{(2\gamma+2\varepsilon)}$ and, as $\varepsilon$ is arbitrary,

Set $x_{0}=x$ and $x_{s}=P_{i_{s}}P_{i_{s-1}}\cdots P_{i_{1}}x$ for $s\geq 1$. Suppose that

Therefore (4.1) holds for every $s$, and we obtain

The implication ”(1′) $\Rightarrow(1)$“ is clear. Note also that the above proof for $k=N$ and $i_{s}=s$ implies that

”(1′) $\Rightarrow$ (6′)“ Note that $\|P_{i_{k}}\cdots P_{i_{1}}-P_{M}\|\leq 1$ always. Suppose now that $a:=\|P_{i_{k}}\cdots P_{i_{1}}-P_{M}\|<1$. We want to show that the range of $I-P_{i_{k}}\cdots P_{i_{1}}$ is closed. Notice first that $\operatorname{Ker}(I-P_{i_{k}}\cdots P_{i_{1}})=M$ since $\{i_{1},\dots,i_{k}\}=\{1,2,\ldots,N\}$. Let $x\in H$ be such that $\operatorname{dist}(x,M)=\|x-P_{M}x\|=1$. We have

Therefore the reduced minimum modulus of $I-P_{i_{k}}\cdots P_{i_{1}})$ verifies $\gamma(I-P_{i_{k}}\cdots P_{i_{1}})\geq 1-\|P_{i_{k}}\cdots P_{i_{1}}-P_{M}\|$. In particular, $\operatorname{Ran}(I-P_{i_{k}}\cdots P_{i_{1}})$ is closed if $a<1$.

The implication ”(6′) $\Rightarrow(6)$“ is easy.

Let $x\in H$, and set $u_{j}=P_{j}\cdots P_{1}x-P_{M}x$ for $j\geq 1$, $u_{0}=x-P_{M}x$. For every $j$ with $1\leq j\leq N$ we have

”$(6)\Rightarrow(3)$“ Let $j$ between $1$ and $N$. Using the Cauchy-Schwarz inequality and Lemma 4.2 we obtain

”(1) $\Rightarrow$ (9)“ Let $\varepsilon>0$. Let $K\subset M^{\perp}$ be a closed subspace of finite codimension in $M^{\perp}$. With respect to the decomposition $H=M\oplus M^{\perp}$, the operator $T$ has the following matrix decomposition

Since $\operatorname{Ran}(T-I)$ is not closed, the range of the operator $I-A$, acting on $M^{\perp}$, is not closed. This means that $I-A\in\mathcal{B}(M^{\perp})$ is not an upper semi-Fredholm operator, and therefore there exists $x\in K$ such that $\|x\|=1$ and $\|x-Ax\|\leq\varepsilon$. It follows that $\|x-Tx\|\leq\varepsilon$.

”(9) $\Rightarrow$ (4)“ Let $x$ be as in (9). Then $x\in K$, $\|x\|=1$, and $\|x-Tx\|\leq\varepsilon$. We have

Set $x_{s}=P_{s}P_{s-1}\cdots P_{1}x$ for $s\geq 1$ and $x_{0}=x$. Then $x_{s}\in M_{s}\cap M^{\perp}$ for each $s\geq 0$ and $x_{s-1}-x_{s}=(I-P_{s})x_{s-1}$ is orthogonal to $x_{s}$. Hence

and $\|x_{s-1}-x_{s}\|\leq\sqrt{2\varepsilon}$, for each $s$. We obtain

For $s\geq 1$ we have $\operatorname{dist}(x,M_{s})\leq\|x-P_{s}P_{s-1}\cdots P_{1}x\|$; hence

Therefore $\max\{\operatorname{dist}(x,M_{j}):j=1,\cdots,N\}\leq N\sqrt{2\varepsilon}.$ As $\varepsilon>0$ is arbitrary, the proof of this implication is over.

”(4) $\Rightarrow$ (9′)“ Suppose that (4) holds. Let $\varepsilon>0$ and let $K\subset M^{\perp}$ be a closed subspace of finite codimension in $M^{\perp}$. Then there exists $x\in K$ such that $\operatorname{dist}(x,M)=\|x\|=1$ and $\max\{\operatorname{dist}(x,M_{j}):j=1,\cdots,N\}\leq\varepsilon$. Let $i_{1},\cdots,i_{k}\in\{1,2,\ldots,N\}$. Set $x_{0}=x$, $x_{s}=P_{i_{s}}\cdots P_{i_{1}}x$ for $s\geq 1$. Then $x_{0}\in K$ and $x_{s}\in M^{\perp}\cap M_{i_{s}}$ for $s\geq 1$.

We shall prove by induction the following two claims :

Both claims are clearly true for $s=0$. Suppose that both inequalities are true for some $s\geq 0$. Then, using several times the induction hypothesis, we have

Thus both (* ‣ 4A) and (** ‣ 4A) are true ; in particular we have

As $\varepsilon>0$ was arbitrary, we obtain (9′).

”(9′) $\Rightarrow$ (8′)“ We have $P_{s}P_{s-1}\cdots P_{1}-P_{M}=P_{s}P_{s-1}\cdots P_{1}(I-P_{M})$, so the range of this operator is in $M^{\perp}$. The assertion (9′) implies that $1$ belongs to the essential spectrum of the restriction of $P_{s}P_{s-1}\cdots P_{1}-P_{M}$ to $M^{\perp}$. Therefore $1\in\sigma_{e}(P_{s}P_{s-1}\cdots P_{1}-P_{M})$.

The implication ”(8′) $\Rightarrow$ (8)“ is clear.

”(8) $\Rightarrow$ (7) $\Rightarrow$ (6)“ The statement (8) implies the following sequence of inequalities for the essential spectral radius $r_{e}(T-P_{M})$ and the essential norm of $T-P_{M}$:

The proofs of implications ”(8′) $\Rightarrow$ (7′) $\Rightarrow$ (6′)“ are similar. The implications ”(8′) $\Rightarrow$ (5′) $\Rightarrow$ (5)“ are clear.

”(9′) $\Rightarrow$ (2′)“ Let $A_{k}$ be the operator $P_{i_{k}}P_{i_{k-1}}\cdots P_{i_{1}}-P_{M}$ restricted to $M^{\perp}$. The condition (9′) implies that $1$ is in the boundary of the essential spectrum of the operator $A_{k}$. According to , on the space $M^{\perp}$ the operators $A_{k}$ converge weakly to $0$. The assertion (2′) can be proved exactly as in [4, Theorem 1] by replacing there $T^{n}$ by $A_{n}$.

The implication ”(2′) $\Rightarrow$ (2)“ is clear.

The implication ”(11) $\Leftrightarrow$ (3)“ follows from . The proof is complete. ∎

B. Quantitative statements

Some remarks concerning the proof of Theorem 4.1 are in order.

Let $\varepsilon>0$. There exists $x\in H$ with $\|x\|=1$ such that $\|P_{i_{k}}\cdots P_{i_{1}}x-P_{M}x\|>a-\varepsilon$. Denote $y=x-P_{M}x$ and

Since $x_{s-1}-x_{s}=(I-P_{i_{s}})x_{s-1}$ is orthogonal to $M_{i_{s}}$, and $x_{s}\in M_{i_{s}}$, we have