Edelstein's Astonishing Affine Isometry
Heinz H. Bauschke, Sylvain Gretchko, Walaa M. Moursi, Matthew Saurette
aa r X i v : . [ m a t h . F A ] S e p Mathematical Assoc. of America American Mathematical Monthly 121:1 September 17, 2020 12:51 a.m. monthly-template200912.tex page 1
Edelstein’s Astonishing Affine Isometry
Heinz H. Bauschke, Sylvain Gretchko,Walaa M. Moursi, and Matthew Saurette
Abstract.
In 1964, Michael Edelstein presented an amazing affine isometry acting on the spaceof square-summable sequences. This operator has no fixed points, but a suborbit that convergesto while another escapes in norm to infinity! We revisit, extend and sharpen his construction.Moreover, we sketch a connection to modern optimization and monotone operator theory.
1. INTRODUCTION.
Suppose that X is a real Hilbert space,with inner product h· , ·i and induced norm k · k . Assume that T : X → X . If thereexists ≤ κ < and k T x − T y k ≤ κ k x − y k , then the celebrated Banach contrac-tion mapping principle guarantees that T has a unique fixed point ¯ x = T ¯ x and that nomatter how the starting point x ∈ X is chosen, the sequence of iterates ( T n x ) n ∈ N converges to ¯ x . But what is the situation when T is merely nonexpansive , i.e., k T x − T y k ≤ k x − y k for all x, y in X ? Well, for starters, T need not have a fixed point (consider trans-lations). Or, T could have many fixed points (consider Id , the identity). And even if T has fixed points, the iterates do not necessarily converge to a fixed point (consider − Id ). We refer the reader to [
3, 7, 8 ] for more on this subject. While these negativeexamples suggest complications, there is indeed more that can be said and also a veryinteresting history!It starts with Browder and Petryshyn who proved in 1966 [ , Theorem 1] that Fix T = ∅ ⇔ for every/some x ∈ X , the sequence ( T n x ) n ∈ N is bounded.(If one orbit is bounded, then so are all by the nonexpansiveness of T .) Negating this,we obtain Fix T = ∅ ⇔ for every/some x ∈ X , the sequence ( k T n x k ) n ∈ N is unbounded.The latter situation was carefully examined by Pazy in 1971 in [ ]. That paper con-tains many fine results on the topic; however, it also states the stronger result that Fix T = ∅ if and only if for every/some x ∈ X , k T n x k → ∞ ; however, the proofis not convincing . Indeed, in a later paper from 1977, Pazy states (see [ , end ofSection 1]) that he does not know whether there exists a fixed-point-free nonexpan-sive map with lim n →∞ k T n x k < ∞ . In hindsight, a proof was impossible to obtain.Indeed, in 1964 — even well before Pazy’s first paper appeared — Edelstein in [ ]constructed a fixed-point free affine isometry for which a subsequence converges evento ! On the positive side, Roehrig and Sine did prove in 1981 (see [ , Theorem 2])that an Edelstein-like example is impossible in finite-dimensional Hilbert spaces, i.e.,Euclidean spaces:January 2014] EDELSTEIN’S ASTONISHING AFFINE ISOMETRY athematical Assoc. of America American Mathematical Monthly 121:1 September 17, 2020 12:51 a.m. monthly-template200912.tex page 2 Fix T = ∅ ⇔ k T n x k → ∞ for every/some x ∈ X provided X is finite-dimensional. The goal of this paper is to bring the amazing discrete dynamical system discovered byEdelstein to a broader audience. It turns out that the material is essentially accessibleto undergraduate mathematics students. We generalize his result, provide full details,and present a new much smaller subsequence that also “blows up to infinity.” Andlast but not least, we interpret this example through the lens of splitting methods andmonotone operator theory! The paper is organized as follows. In Section 2, we reviewaffine rotations which form the basic building blocks of Edelstein’s example. Theserotations are lifted to ℓ in Section 3. Section 4 deals with estimating the norm of theorbit starting at . A suborbit converging to is presented in Section 5, while Section 6provides a suborbit blowing up in norm to ∞ ! In Section 7, we sketch a connection tothe Douglas–Rachford splitting operator. We conclude with an epilogue in Section 8.
2. AFFINE ROTATIONS IN R . We start with the linear rotation matrix and a fixedvector, i.e., L := L θ := (cid:18) cos θ − sin θ sin θ cos θ (cid:19) and v = (cid:18) v v (cid:19) . (1)The affine rotation operator we will first investigate is Rx := R θ x := L θ x + v. (2)Note that R is an affine isometry (also known as isometric affine mapping ), i.e., k Rx − Ry k = k x − y k for all x and y in R . (3)(Also, L is a linear isometry.) Let’s determine the fixed points of R : for x ∈ R , wehave x ∈ Fix R ⇔ x = Rx ⇔ x = Lx + v ⇔ (Id − L ) x = v . (4)Now Id − L = (cid:18) − cos θ sin θ − sin θ − cos θ (cid:19) has determinant (1 − cos θ ) + sin θ = 2(1 − cos θ ) . Assuming that cos θ < , thematrix Id − L is invertible with (Id − L ) − = 12(1 − cos θ ) (cid:18) − cos θ − sin θ sin θ − cos θ (cid:19) = 12 (cid:18) − cot( θ/ θ/
2) 1 (cid:19) ; thus, also using (4), f := (Id − L ) − v = 12(1 − cos θ ) (cid:18) (1 − cos θ ) v − (sin θ ) v (sin θ ) v + (1 − cos θ ) v (cid:19) (5a)2 c (cid:13) THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 121 athematical Assoc. of America American Mathematical Monthly 121:1 September 17, 2020 12:51 a.m. monthly-template200912.tex page 3 = 12 (cid:18) v − v cot( θ/ v + v cot( θ/ (cid:19) (5b)is the unique fixed point of R and f − Lf = v. (If cos θ = 1 , then Rx = x + v and thus either (i) v = 0 and Fix R = R or (ii) v = 0 and Fix R = ∅ .) It is now easily shown by induction that L nθ = L nθ and that R n x = f + L nθ ( x − f ) for every n ∈ N . (6)Let us specialize this further by setting v = (cid:18) v v (cid:19) = ξ · (cid:18) − cos θ − sin θ (cid:19) , (7)where ξ > is a parameter. (From now on, we can again include the case when cos θ = 1 !) With this particular assignment, the fixed point f of R found in (5) sim-plifies to f = (cid:18) ξ (cid:19) . (8)It follows from (6) and (1) that R n x = (cid:18) ξ + ( x − ξ ) cos( nθ ) − x sin( nθ )( x − ξ ) sin( nθ ) + x cos( nθ ) (cid:19) ; in particular, R n ξ · (cid:18) − cos( nθ ) − sin( nθ ) (cid:19) . Using the half-angle formula for the squared sine, we obtain k R n k = ξ (cid:0) − nθ ) (cid:1) = 4 ξ sin ( nθ/ . (9)More generally, tackling R n x , we have k R n x k = k x k + 2 ξx (cos( nθ ) − − ξx sin( nθ ) + 2 ξ (1 − cos( nθ ))= k x k − ξx sin ( nθ/ − ξx sin( nθ/
2) cos( nθ/
2) + 4 ξ sin ( nθ/ . Soon, we will “lift” R from R to ℓ . To do so, we develop some bounds. For α and β in R , we clearly have − ( α + β ) ≤ αβ ≤ ( α + β ) ; thus, − α + β ) ≤− αβ ≤ α + β ) . Hence − ξx sin ( nθ/ − ξx sin( nθ/
2) cos( nθ/ ≤ x + ξ sin ( nθ/ x + ξ sin ( nθ/
2) cos ( nθ/ January 2014]
EDELSTEIN’S ASTONISHING AFFINE ISOMETRY athematical Assoc. of America American Mathematical Monthly 121:1 September 17, 2020 12:51 a.m. monthly-template200912.tex page 4 = 2 k x k + 2 ξ sin ( nθ/ ( nθ/
2) + cos ( nθ/ k x k + 2 ξ sin ( nθ/ . Similarly, − ξx sin ( nθ/ − ξx sin( nθ/
2) cos( nθ/ ≥ − k x k − ξ sin ( nθ/ . Altogether, we finally have − k x k + 2 ξ sin ( nθ/ ≤ k R n x k ≤ k x k + 6 ξ sin ( nθ/ . (10)
3. FROM R TO ℓ : LIFTING R TO R . From now on, ℓ is the real Hilbert spaceof all square summable sequences. We think of ℓ here as the subset ℓ ( R × R × · · · and we will think of the k th plane as indexed by k ∈ { , , . . . } . Now consider a sequence of angles ( θ k ) k ≥ , where θ k is the angle for k th plane, as well as a sequence of positive parameters ( ξ k ) k ≥ . Set, in the spirit of (7), v k = ξ k · (cid:18) − cos( θ k ) − sin( θ k ) (cid:19) ∈ R for k ≥ . Using (9), we estimate k v k k = 4 ξ k sin ( θ k / ≤ min { ξ k , ξ k θ k } . (11)Then v := ( v k ) k ≥ will lie in ℓ , e.g., if ξ := ( ξ k ) k ≥ lies in ℓ or if ξ is bounded and θ := ( θ k ) k ≥ ∈ ℓ . An ingenious choice by Edelstein [ ] will lead to a nice analysis:we set θ k = 2 πk ! ∈ (0 , π ] , (12)which gives k v k = k ( v k ) k ≥ k = 4 X k ≥ ξ k sin ( π/k !) . (13)We assume from now on that ( ξ k ) k ≥ is positive and decreasingbut not necessarily strictly decreasing (indeed, Edelstein used ξ k ≡ ). Using (11), wehave k v k ≤ X k ≥ ξ k π ( k !) < π ξ X k ≥ k ! = 4 π ξ (exp(1) − . (14)We are now ready to extend R to the countable Cartesian product of Euclidean planes:Set R : x = ( x , x , . . . , x k , . . . ) ( R θ x , R θ x , . . . , R θ k x k , . . . ) , (cid:13) THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 121 athematical Assoc. of America American Mathematical Monthly 121:1 September 17, 2020 12:51 a.m. monthly-template200912.tex page 5 where each x k ∈ R . From (10), (13) and (14), we have, for x ∈ ℓ , k Rx k ≤ k x k + 6 X k ξ k sin ( π/k !) = 3 k x k + (3 / k v k < ∞ (15)and thus R is an affine operator from ℓ to itself! Owing to (3), we have that k Rx − Ry k = k x − y k for all x and y in ℓ .What about the corresponding fixed point? Well, in view of (8), an (algebraic) fixedpoint of R is f := ( ξ , , ξ , , ξ , , . . . ) and k f k = X k ≥ ξ k ∈ [0 , + ∞ ] . So the original Edelstein choice ξ k ≡ leads to f / ∈ ℓ , i.e., Fix R = ∅ provided that ξ k ≡ .(Other choices with the same outcome are possible, e.g., consider ξ k ≡ / √ k .)
4. ESTIMATING k R N (0) k . Observe that (10) yields k R n x k ≤ k x k + 6 X k ≥ ξ k sin ( nπ/k !) , while (9) gives the exact identity k R n k = 4 X k ≥ ξ k sin ( nπ/k !) . (16)For numerical computations, we need to estimate the convergence of this infinite se-ries. Indeed, ≤ k R n k − n X k =1 ξ k sin ( πn/k !) = 4 X k ≥ n +1 ξ k sin ( πn/k !) ≤ π ξ X k ≥ n +1 ( n/k !) ≤ π ξ (cid:18) n + 1) + 1( n + 1) ( n + 2) + 1( n + 1) ( n + 2) ( n + 3) + · · · (cid:19) < π ξ (cid:18) n + 1) + 1( n + 1) + 1( n + 1) + · · · (cid:19) = 4 π ξ n ( n + 2) January 2014]
EDELSTEIN’S ASTONISHING AFFINE ISOMETRY athematical Assoc. of America American Mathematical Monthly 121:1 September 17, 2020 12:51 a.m. monthly-template200912.tex page 6 Figure 1. k R n k for n ≤ . Figure 2. k R n k for n ≤ . → as n → ∞ .Hence we will use for numerical computation k R n k ≈ n X k =1 ξ k sin ( πn/k !) . We present k R n k for the first 250 (respectively, 1000) iterates in Figure 1 (respec-tively, Figure 2). The source code is available at [ ].Can you predict the long-term behavior from these plots? We certainly would haveguessed some form of periodic behavior. However, this guess is far from the truth as we6 c (cid:13) THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 121 athematical Assoc. of America American Mathematical Monthly 121:1 September 17, 2020 12:51 a.m. monthly-template200912.tex page 7 will see in the following sections. Indeed, a suborbit will converge to while anotherwill blow up in norm to ∞ . This bizarre behavior is not at all obvious from the plots!
5. A SUBORBIT THAT CONVERGES TO . For any n ≥ , we have k R n ! k = 4 X k ≥ ξ k sin ( πn ! /k !) = 4 X k ≥ n +1 ξ k sin ( πn ! /k !) ≤ π ξ X k ≥ n +1 ( n ! /k !) ≤ π ξ (cid:18) n + 1) + 1( n + 1) ( n + 2) + 1( n + 1) ( n + 2) ( n + 3) + · · · (cid:19) < π ξ (cid:18) n + 1) + 1( n + 1) + 1( n + 1) + · · · (cid:19) = 4 π ξ n ( n + 2) → as n → ∞ .Hence, we have the wonderfully weird suborbit result, discovered first by Edelstein: Theorem 1. R n ! (0) → ; in fact, k R n ! (0) k ≤ O (cid:0) n (cid:1) . This is very surprising and certainly not something easily deduced from the plots.(Of course, n ! grows really fast to + ∞ .) However, in Figure 1 and Figure 2 we canindeed make out small values, namely for k R k and for k R k . This is not acoincidence since
120 = 5! and
720 = 6! .In the next section, we will reveal a new suborbit that blows up to infinity.
6. A NEW SUBORBIT THAT BLOWS UP TO ∞ . Edelstein already provided asuborbit that goes to ∞ in norm. Indeed, he suggested the sequence of indices e n := n X m =1 ( n m )! for which he proved that k R e n k → ∞ . It is hard to grasp just how fast ( e n ) n ≥ grows. Indeed, the first three terms are ( e , e , e ) = (1 , , . We won’t list e , e , e but point out that they have , , decimal digits, re-spectively. (The source code is available at [ ].) Instead, we present a new specialsubsequence that will achieve the same result but with much smaller indices. Set s n := 1 + n − X m =1 ⌈ m/ ⌉ ( m + 2)! January 2014]
EDELSTEIN’S ASTONISHING AFFINE ISOMETRY athematical Assoc. of America American Mathematical Monthly 121:1 September 17, 2020 12:51 a.m. monthly-template200912.tex page 8 for every n ≥ . Of course, s n approaches infinity but much slower than ( e n ) n ≥ ;indeed, ( s , s , s , s , s , s ) = (7 , , , , , . We now start to analyze the term s n . First, s n k ! = 1 k ! + n − X m =1 ⌈ m/ ⌉ ( m + 2)! /k != 1 k ! + k − X m =1 ⌈ m/ ⌉ ( m + 2)! /k ! + n − X m = k − ⌈ m/ ⌉ ( m + 2)! /k ! | {z } ∈ Z , where k − ≤ n − , i.e., k ≤ n + 2 . Next, we will work on deriving bounds onthe fractional part of s n /k ! which will be used later. Assuming that k − ≥ , i.e., k ≥ , we have k ! + k − X m =1 ⌈ m/ ⌉ ( m + 2)! /k ! > k − X m = k − ⌈ m/ ⌉ ( m + 2)! /k != ⌈ ( k − / ⌉ /k ≥ ( k − / (2 k )= 12 − k . We now turn to an upper bound. Using ⌈ q/ ⌉ < ( q + 2) / and requiring that k ≥ ,we obtain k − X m =1 ⌈ m/ ⌉ ( m + 2)! k != ⌈ ( k − / ⌉ ( k − k ! + ⌈ ( k − / ⌉ ( k − k ! + k − X m =1 ⌈ m/ ⌉ ( m + 2)! k != ⌈ ( k − / ⌉ k + ⌈ ( k − / ⌉ k ( k −
1) + k − X m =1 ⌈ m/ ⌉ ( m + 3) · · · ( k − k − k< k − k + k − k ( k −
1) + k − X m =1 ( m + 2) / k − k − k = ( k + 1)(2 k − k + 4)4( k − k − k . It follows that k ! + k − X m =1 ⌈ m/ ⌉ ( m + 2)! k ! < k ! + ( k + 1)(2 k − k + 4)4( k − k − k< k ! + ( k + 1)(2 k − k + 4)4( k − k − k (cid:13) THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 121 athematical Assoc. of America American Mathematical Monthly 121:1 September 17, 2020 12:51 a.m. monthly-template200912.tex page 9 = 1 k ! + k + 12 k = 12 + 1 k ! + 12 k<
12 + 1( k )(4)(3)(2) + 12 k = 12 + 1324 k . Altogether, the fractional part { s n /k ! } of s n /k ! satisfies, for k ≥ , <
720 = 12 − ≤ − k < n s n k ! o <
12 + 1324 k ≤
12 + 13240 = 133240 < . Combining this with the fact that sin over [ π/ , π/ has minimum value / re-sults in < sin (cid:16) s n k ! π (cid:17) whenever ≤ k ≤ n + 2 .Using (16), we thus get k R s n k = 4 ∞ X k =1 ξ k sin ( s n π/k !) > n +2 X k =10 ξ k ≥ n − ξ n +2 . (17)We now turn toward an upper bound. Again using ⌈ q/ ⌉ < ( q + 2) / as well asan easy induction in (18b), we obtain s n = 1 + n − X m =1 ⌈ m/ ⌉ ( m + 2)! < n − X m =1 12 ( m + 2)( m + 2)! (18a) = 1 + ( n + 2)! − (18b) < ( n + 2)! (18c)which implies s n ( n + 3)! < n + 3) , s n ( n + 4)! < n + 3)( n + 4) ,s n ( n + 5)! < n + 3)( n + 4)( n + 5) , . . . . Thus ∞ X k = n +3 ξ k sin ( s n π/k !) < π ∞ X k = n +3 ξ k ( n + 3) ( n + 4) · · · k < π ξ n +3 ( n + 3) (cid:18) n + 3) + 1( n + 3) + · · · (cid:19) = π ξ n +3 ( n + 3) − . January 2014]
EDELSTEIN’S ASTONISHING AFFINE ISOMETRY athematical Assoc. of America American Mathematical Monthly 121:1 September 17, 2020 12:51 a.m. monthly-template200912.tex page 10 Again using (16), we thus have k R s n k = 4 ∞ X k =1 ξ k sin ( s n π/k !) < n +2 X k =1 ξ k + π ξ n +3 ( n + 3) − . (19)Combining (17) with (19), we obtain the following result, which summarizes ourwork in this section: Theorem 2. If n ≥ , then n − ξ n +2 ≤ n +2 X k =10 ξ k ≤ k R s n k < n +2 X k =1 ξ k + π ξ n +3 ( n + 3) − ≤ n + 2) ξ + π ξ n +3 ( n + 3) − . Consequently, we have the implications lim k →∞ ξ k > ⇒ ( ξ k ) k ≥ / ∈ ℓ ⇒ k R s n k → ∞ . and lim k →∞ ξ k > ⇒ k R s n k = O (cid:0) √ n (cid:1) . Note that this covers the Edelstein set-up where ξ k ≡ , albeit with the muchsmaller sequence of indices ( s n ) n ≥ compared to ( e n ) n ≥ .
7. OPTIMIZATION AND THE DOUGLAS–RACHFORD ALGORITHM.
Inthis section, we place the example into a different framework. Indeed, we have so farstudied the world of nonexpansive mappings. There are two related worlds, namelythe ones featuring firmly nonexpansive mappings and monotone operators. We startwith firmly nonexpansive mappings, which are mappings on X that satisfy k T x − T y k ≤ h T x − T y, x − y i for all x and y in X .It is not hard to show that T is firmly nonexpansive if and only if T − Id is nonex-pansive, which implies that the firmly nonexpansive mappings are precisely those thatcan be written as the average of the identity and nonexpansive mappings (see, e.g., [ ]for more on this).Let’s identify the firmly nonexpansive counterpart of the operator R = R θ consid-ered in (2), with < θ ≤ π . We thus set T := T θ := Id + R θ . Let x ∈ X . Using the notation and results of Section 2, it can be shown by inductionthat T n x = f + 12 n (Id + L ) n ( x − f ) .
10 c (cid:13)
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Using the double-angle formula for sine and half-angle formula for cosine (with angle θ/ ), we obtain Id + L θ = 2 cos( θ/ L θ/ . Hence (Id + L θ ) / θ/ L θ/ andthus (cid:0) (Id + L θ ) / (cid:1) n = cos n ( θ/ L nθ/ . It follows that if θ < π , then T n x = f + cos n ( θ/ L nθ/ ( x − f ) → f (20)as predicted by Rockafellar’s proximal point algorithm (see, e.g., [ ]); indeed, k T n x − f k = cos n ( θ/ k x − f k → and T n x → f with a sharp linear rate of cos( θ/ . And if θ = 2 π , then T π = Id andwe have immediate and even finite convergence to a fixed point!The operator T on R gives rise to an induced operator T : ℓ → ℓ (because R is well-defined on ℓ by (15) and hence so is T = (Id + R ) / ). While T does havethe unique fixed point f (unless θ = 2 π ) the same is no longer true in general for T because an algebraic fixed point f may fail to lie in ℓ . ( Fix T cannot be a singletonbecause Fix T θ = Fix T π = R .)Let us now interpret this from an optimization/feasibility perspective. Let U and V be nonempty closed convex subsets of R with U ∩ V = ∅ , and denote the projectionmappings onto U, V by P U , P V , respectively. Then the associated Douglas–Rachfordsplitting operator is D := D V,U = Id − P U + P V (2 P U − Id) . It is known that the sequence ( P U D n x ) n ≥ converges to some point in U ∩ V ; thisis a well-known method to solve feasibility (and even optimization) problems. Nowsuppose that U = R (1 ,
0) = R × { } and V = f + R (cos( θ/ , sin( θ/ , where f = ( ξ, is as in (8). Clearly, U ∩ V = { f } (unless θ = 2 π , in which case U ∩ V = U = V ). Using linear algebra, one may check that D = T ; put differently, the Douglas–Rachford operator for the feasibility problem of finding apoint in U ∩ V is exactly the firmly nonexpansive counterpart of the Edelstein affineisometry! Working towards the ℓ version, consider first the pure Cartesian products e U = ( R × { } ) × ( R × { } ) × · · · and e V = (cid:0) ( ξ ,
0) + R (cos( θ / , sin( θ / (cid:1) × (cid:0) ξ ,
0) + R (cos( θ / , sin( θ / (cid:1) × · · · = (cid:0) ( ξ ,
0) + R (cos( π/ , sin( π/ (cid:1) × (cid:0) ξ ,
0) + R (cos( π/ , sin( π/ (cid:1) × · · · = (cid:0) R × { } (cid:1) × (cid:0) { ξ } × R ) (cid:1) × · · · × (cid:0) ( ξ k ,
0) + R (cos( π/k !) , sin( π/k !)) (cid:1) × · · · January 2014]
EDELSTEIN’S ASTONISHING AFFINE ISOMETRY athematical Assoc. of America American Mathematical Monthly 121:1 September 17, 2020 12:51 a.m. monthly-template200912.tex page 12 for which e U ∩ e V = (cid:0) R × { } ) × { ( ξ , } × · · · × { ( ξ k , } × · · · . Now set U = e U ∩ ℓ and V = e V ∩ ℓ . Then our T is precisely the Douglas–Rachford operator for finding a point in U ∩ V .Note that this set is possibly empty when ξ / ∈ ℓ , as is the case for Edelstein’s original ξ k ≡ .We finally turn to the third world, the world of (maximally) monotone operators.The unique maximally monotone operator associated with R and with T is M := M θ := T − θ − Id . Let us find out what M is. We start by inverting T , for which (20) surely helps. So wewrite x = T y . Let us abbreviate c := cos( θ/ and K := L θ/ . Assume that c = 0 ,i.e., θ = π . Note that K − = L − θ/ = L ∗ θ/ = K ∗ . Then x = T y = f + cK ( y − f ) if and only if c − ( x − f ) = K ( y − f ) if and only if c − K − ( x − f ) = y − f if and only if f + c − K ∗ ( x − f ) = y . Hence M x = f + c − K ∗ ( x − f ) − x =( c − K ∗ − Id)( x − f ) . Switching back to the original notation, we have M x = (cid:18) θ/ (cid:18) cos( − θ/ − sin( − θ/ − θ/
2) cos( − θ/ (cid:19) − (cid:18) (cid:19)(cid:19) ( x − f )= (cid:18) θ/ − tan( θ/
2) 0 (cid:19) ( x − f )= tan( θ/ (cid:18) − (cid:19) ( x − f ) . If θ = π , then T x ≡ f and hence the set-valued inverse of T maps f to R andanything else to the empty set. To summarize, M θ x = tan( θ/ − ! ( x − f ) , if θ = π ; R , if θ = π and x = f ; ∅ , if θ = π and x = f . (21)Monotone operator theory predicts that M is monotone , i.e., h x ∗ − y ∗ , x − y i ≥ forall x ∗ ∈ M x and y ∗ ∈ M y . Indeed, our operator M satisfies this inequality even asequality.Recall that the Edelstein angles given by (12) are θ = 2 π , θ = π , θ = π/ , . . . , θ k = 2 π/k ! , . . . . Let k ≥ . Then θ k = π . Writing x = ( x , x ) , t = tan( θ k / ,and ξ = ξ k , it follows from (21) and (8) that M x = t ( − x , x − ξ ) and thus k M x k = t ( x + ( x − ξ ) ) = t ( k x k + ξ − ξx ) ≤ t ( k x k + 2 ξ + x ) ≤ t ( k x k + ξ ) .
12 c (cid:13)
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Switching to the product space version M acting on R × R × · · · , we deduce thatif u = ( u k ) k ≥ ∈ Mx and x ∈ ℓ , then k u k − k u k − k u k = X k ≥ k u k k ≤ X k ≥ tan ( θ k / k x k k + ξ k ) ≤ k x k + ξ ) X k ≥ tan ( π/k !) < ∞ because tan ( π/k !) = (sin( π/k !) / cos( π/k !)) ≤ ( π/k !) / cos ( π/ π / ( √ / / ( k !) = (4 π / / ( k !) and the comparison test applies. Hence for x = ( x , x , . . . ) ∈ ℓ $ R × R × . . . we have Mx = ∅ ⇔ x = ( ξ , . This concludes our journey featuring the Edelstein operator — we hope you enjoyedthe ride as much as we did!
8. EPILOGUE.
The authors believe that Jonathan (Jon) Borwein would have likedthe material in this paper for several reasons: While at Dalhousie University, Jon andMichael Edelstein were actually colleagues. Jon enjoyed nonexpansive mappings andpublished extensively in this area (see, e.g., [ ]). In his optimization work, Jon spentsignificant time of his last years working on the Douglas–Rachford algorithm (see,e.g., [ ]). Last but not least, some of the numbers in this paper were obtained by com-putation (see [ ]) — as a co-founder of experimental mathematics (see, e.g., [ ]), Jonwould have enjoyed the flavor of these results and the concreteness of the examples. ACKNOWLEDGMENT.
The authors thank the referees and the editors for their careful reading and con-structive comments. HHB was partially supported by the Natural Sciences and Engineering Research Councilof Canada. WMM was partially supported by a Natural Sciences and Engineering Research Council of CanadaPostdoctoral Fellowship.
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HEINZ BAUSCHKE is a former doctoral student of Jonathan Borwein and currently a Professor of Mathe-matics at the University of British Columbia (Okanagan campus in Kelowna, B.C., Canada). His main areas ofinterest are in convex analysis, optimization, and monotone operator theory. He has published more than 100papers and the book
Convex Analysis and Monotone Operator Theory in Hilbert Spaces . Department of Mathematics, UBC Okanagan, Kelowna, B.C. V1V 1V7, [email protected]
SYLVAIN GRETCHKO is a Software Engineer and a graduate student in mathematics at the University ofG¨ottingen. He is particularly interested in experimental mathematics.
Faculty of Mathematics and Computer Science, University of G¨ottingen, 37073 G¨ottingen, [email protected]
WALAA MOURSI is an Assistant Professor in the Department of Combinatorics and Optimization at theUniversity of Waterloo. Her research interests are convex analysis, monotone operator theory and continuousoptimization.
Department of Combinatorics and Optimization, Faculty of Mathematics, University of Waterloo, Waterloo,Ontario N2L 3G1, [email protected]
MATTHEW SAURETTE is an undergraduate student in mathematics at the University of British Columbia(Okanagan campus in Kelowna, B.C., Canada). His interests are in optimization and mathematical biology.
Department of Mathematics, UBC Okanagan, Kelowna, B.C. V1V 1V7, [email protected]
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