The Blum-Hanson Property

Abstract Given a (real or complex, separable) Banach space, and a contraction T on X, we say that T has the Blum-Hanson property if whenever x, y ∈ X are such that Tnx tends weakly to y in X as n tends to infinity, the means 1N∑k=1NTnkx {1 \over N}\sum\limits_{k = 1}^N {{T^{{n_k}}}x} tend to y in norm for every strictly increasing sequence (nk) k≥1 of integers. The space X itself has the Blum-Hanson property if every contraction on X has the Blum-Hanson property. We explain the ergodic-theoretic motivation for the Blum-Hanson property, prove that Hilbert spaces have the Blum-Hanson property, and then present a recent criterion of a geometric flavor, due to Lefèvre-Matheron-Primot, which allows to retrieve essentially all the known examples of spaces with the Blum-Hanson property. Lastly, following Lefèvre-Matheron, we characterize the compact metric spaces K such that the space C(K) has the Blum-Hanson property.


Introduction
These notes present the material for a mini-course on the Blum-Hanson property, given within the framework of the ACOTCA 2019 conference at the Université Paris-Est in Marne-la-Vallée (France) in June 2019. They were written down by Clément Coine. The mini-course consisted of three lectures of 45 minutes. The structure of this course is preserved in these notes, and the contents of the three lectures correspond to the contents of Sections 2, 3 and 4 respectively. means that xn tends to x in norm in X as n tends to in nity, while the notation xn w − → x means that xn tends weakly to x in X. We denote by B(X) the algebra of bounded operators on X, by B X the closed unit ball of X, and by S X its unit sphere. Theorem 1. [3,11] Let H be a (real or complex) Hilbert space, and let T ∈ B(H) with T ≤ . If x, y ∈ H are such that T n x w − → y, then N N k= T n k x . − − → y for every strictly increasing sequence (n k ) k≥ of (positive) integers.
This theorem motivates the following de nition: De nition 2. Let X be a (real or complex) Banach space, and let T ∈ B(X). We say that T has the BH property (or simply has BH) if whenever x, y ∈ X are such that T n x w − → y, N N k= T n k x . − − → y for every strictly increasing sequence (n k ) k≥ of integers. We say that X itself has the BH property if every contraction on X has the BH property.
In the rst part of these lectures (Section 2), we will quickly present the theorem of Blum and Hanson and its ergodic-theoretic motivation. We will also prove Theorem 1 and give some examples of spaces which have BH. During the second lecture (Section 3), we will present and prove a recent criterion, due to Lefèvre-Matheron-Primot [15], proving that certain contractions (sometimes all contractions) on certain Banach spaces have BH. We will present some of its applications, as well as its limits. Finally, the last part of this mini-course (Section 4) will be devoted to the study of spaces which do not have the BH property.

Strongly mixing dynamical systems and the BH Theorem
Let (X, B, µ) be a probability space, and let ϕ : X → X be a measure-preserving transformation of (X, B, µ): this means that µ(ϕ − (A)) = µ(A) for every A ∈ B. One associates to ϕ a canonical isometry U ϕ on L (X, B, µ), called the Koopman operator and de ned as follows: When ϕ is an invertible measure-preserving transformation, U ϕ is a unitary operator. For a rst reading in ergodic theory, we recommend the classical book [18] by Walters.
Von Neumann's mean ergodic theorem implies that for every f ∈ L (X, B, µ), where P ker(U ϕ −I) denotes the orthogonal projection on the eigenspace ker(U ϕ − I) of The transformation ϕ is said to be ergodic when the only ϕ-invariant functions f ∈ L (X, B, µ) are constant almost everywhere: f • ϕ = f µ-a.e. ⇒ f = c µ-a.e. Another way of saying this is that ker(U ϕ − I) is -dimensional. In this case, where . , . denotes the scalar product in L (X, B, µ). This is equivalent to the condition and to the condition that if A ∈ B is such that ϕ − (A) = A up to a set of µ-measure , then µ(A) = or µ(A) = .
Ergodic systems are the basic building blocks for all measure-preserving systems (a good illustration of this is given by the Ergodic Decomposition Theorem, see for instance [1, Th. 2.2.9]); they satisfy Birkho 's pointwise ergodic theorem: for every f ∈ L (X, B, µ), which is classically rephrased as "the time means equal the space mean µ-a.e.".
The simplest examples of ergodic systems are the irrational rotations on the unit circle, but there are many more examples, in various contexts (see one of the references [18], [17] or [9]).
Let us go back to the de nition of ergodicity in terms of Koopman operators: for all f , g ∈ L (X, B, µ), There are several natural reinforcements of this notion, where one requires a di erent kind of convergence of the quantities U k ϕ f , g above. One of them is strong mixing: De nition 3. A measure-preserving transformation ϕ of (X, B, µ) is strongly mixing if for every f , g ∈ L (X, B, µ), This is equivalent to the condition µ(ϕ −N (A) ∩ B) → µ(A)µ(B) for every A, B ∈ B, i.e. to the condition that the events ϕ −N (A) and B become asymptotically independent as N goes to in nity. Hence the terminology "strongly mixing".
Rotations of the unit circle are never strongly mixing. But endomorphisms of the tori R n /Z n are strongly mixing as soon as they are ergodic. Endomorphisms of tori are given by n × n matrices with integer entries: to each such matrix . . . a n . . . . . . . . . a n . . . ann     one associates the map Φ A : Here are the simplest examples of endomorphisms of tori: take n = , A = (p) with p ∈ N\ { , }: one obtains the map Endomorphisms of tori preserve Haar measure (for example, check that for every semi-open arc I in R/Z, µ({x ; px mod ∈ I}) = µ(I)). When Φ A is surjective (which happens exactly when det(A) ≠ ), Φ A is ergodic if and only if A has no roots of unity as eigenvalues, if and only if Φ A is strongly mixing. See [18] for details.
Going back to the theory of strongly mixing systems, we observe that ϕ is strongly mixing if and only if Here is the characterization of strongly mixing systems obtained by Blum and Hanson in .
Theorem 4. [8] The (measure-preserving) dynamical system (X, B, µ; ϕ) is strongly mixing if and only if for every strictly increasing sequence (n k ) k≥ of integers, we have One can replace the norm . by any norm . p, ≤ p < +∞, in the statement of Theorem 4. But one cannot replace it by pointwise convergence. An example of a strongly mixing system (X, B, µ; ϕ) for which there exists a strictly increasing sequence (n k ) k≥ of integers and a function f ∈ L (X, B, µ) such that was rst given in [10], and then Krengel proved in [12] that there exists a universal strictly increasing sequence (n k ) k≥ of integers such that for every strongly mixing system (X, B, µ; ϕ), there exists a set A ∈ B with the property that where 1 A denotes the indicator function of A.
As we already mentioned, the theorem of Blum and Hanson admits an abstract formulation valid for all contractions T on a Hilbert space H (Theorem 1 above), also called a mean ergodic theorem along all subse- for every strictly increasing sequence of integers (n k ) k≥ . In the case where n k = k for all k, this is a classical mean ergodic theorem, valid on all Banach spaces X (see for instance [13, Ch.2, Th. 1.1]). In the same circle of ideas, recall that whenever T is a power-bounded operator on a re exive Banach space X (i.e. sup ||T n || < +∞; this holds true in particular when T is a contraction), the averages N N k= T k x converge in norm in X to a vector y belonging to ker(T − I).
Let us now prove Theorem 1.
Proof of Theorem 1. Since Ty = y, we can assume without loss of generality that y = . Thus, we suppose that An important ingredient in all the existing proofs of Theorem 1 is the following fact.
The proof relies on a Cesáro type argument. Let ε > and K ∈ N be such that ≤ c ij < ε for every pair (i, j) of integers with |i − j| ≥ K. We have Once this fact is observed, there are several ways of proving Theorem 1. Probably the most elegant argument is the one presented in [15, Sec. 6.1], which relies on the existence of spectral measures for contractions on complex Hilbert spaces. We prefer to follow here the elementary approach from [3] (see [13, Ch.8, Th. 1.3]), which runs as follows: Note that the sequence ( T n x ) n≥ is decreasing so that the limit lim n→+∞ T n x exists. Hence, given ε > , there exists K ≥ such that, for all k ≥ K and for all i ≥ , One now observes the following fact: if S ∈ B(H) and u ∈ H are such that S ≤ and ≤ u − Su < ε , then | u, y − Su, Sy | ≤ ε y for every y ∈ H. Indeed, Apply this with S = T i and u = T k x, y = x to get 3. p(N), for < p < ∞, has BH, see [16,Th. 2.5]. This is one of the important recent results on the BH property, which motivated the work of Lefèvre-Matheron-Primot [15] which is the object of the next section. 4. By [4], positive contractions on spaces L p (Ω, F, µ), < p < +∞, where (Ω, F, µ) is a standard probability space, have BH. It is unknown whether all contractions on L p (Ω, F, µ) have BH, i.e. whether L p (Ω, F, µ) has BH for < p ≠ < +∞. This is one of the major open questions concerning the BH property, see [5]. 5 We will present in the next lecture a criterion from [15] which allows to retrieve all positive results on the BH property thanks to a rather geometric argument, involving the asymptotic behavior at in nity of a certain "modulus of smoothness".

A geometric criterion for the BH property
Let us begin by xing some notation. Let X be a real, separable Banach space, C ⊂ X a convex cone (that is, C is a non-empty convex set such that t C ⊂ C for every t ≥ ). Let us set In other words, the set WN(B X ∩ C) consists of all weakly null sequences in B X ∩ C. For every x ∈ C and every t ≥ , de ne In the rest of the paper, we will use the following terminology: if x ∈ X and T ∈ B(X) with T ≤ , we say that T satis es the BH property at x if the weak convergence T n x w − → implies that for every strictly increasing sequence (n k ) k≥ of integers, N N k= T n k x −→ N→+∞ . Theorem 6. [15] Suppose that for every x ∈ C, Then every operator T ∈ B(X) with T ≤ and T(C) ⊂ C satis es the BH property at every point x ∈ C.
Here are some straightforward remarks on the functions r C (x, . ).
-the map t → r C (x, t) is -Lipschitz on [ , +∞), so that the function r C (x, t) − t is decreasing and the quantity lim -the function t → r C (x, t) is increasing on [ , +∞).

Some applications and examples:
if one wishes to show, thanks to Theorem 6, that a given space X has the BH property, one applies it to C = X (which is indeed a symmetric convex cone).
1. X = p(N), < p < +∞: in this case, r B X (x, t) = ( x p + t p ) /p for every x ∈ X and every t ≥ . Thus for every x ≠ , and this tends to as t → +∞. Hence X has the BH property.
. So X has the BH property, see [7]. 3. X = L p (Ω, F, P), < p < +∞, where (Ω, F, P) is a standard probability space, for instance the interval [ , ] with Lebesgue measure: as we already mentioned at the end of Section 2, it is unknown whether X has BH for p ≠ . We observe next, following [15,Sec. 6.4] that Theorem 6 does not apply in this case: Proof. Let a, b > with a ≠ b and let λ ∈ ( , ). Let (ξn)n be a sequence of independent random variables on (Ω, F, P) with P(ξn = a) = λ, P(ξn = −b) = − λ, E(ξn) = and ξn p = . The last two conditions place constraints on the parameters a, b and λ. We must have The sequence (ξn)n tends weakly to in L p (Ω, F, P). Indeed, ξn ∈ L (Ω, F, P) for each n, and since the ξn's are independent and satisfy E(ξn) = , they are orthogonal in L (Ω, F, P): E(ξn ξm) = E(ξn)E(ξm) = if m ≠ n. Moreover, ξn ∞ ≤ max(a, b), so the sequence (ξn)n is bounded in L (Ω, F, P), and thus ξn w − → in L (Ω, F, P). An approximation argument, using the fact that sup n ξn ∞ < +∞, then shows that ξn w − → in L p (Ω, F, P). Now, The fact that λa p + ( − λ)b p = and straightforward computations show that Since λa = ( − λ)b, the right-hand side is equal to λa(a p− − b p− ). If p = , this term is equal to and if p ≠ , the parameters can be chosen in such a way that this term is positive (take a < b if p < and a > b if p > ).
On the other hand, Theorem 6 can be applied to show that positive contractions on L p (Ω, F, P) have the BH property at every positive f ∈ L p (Ω, F, P): we apply the theorem to C = L p + (Ω, F, P), where L p + (Ω, F, P) = {f ∈ L p (Ω, F, P) ; f ≥ a.e. on Ω}.
Let (fn)n ⊂ B L p ∩ C, fn w − → : since fn ≥ for every n, (fn) converges in probability to , i.e. P(|fn| > ε) → for every ε > . Hence, for any f ∈ L p (Ω, F, P),  (1). This inequality means that the supports of f and fn become asymptotically disjoint as n goes to in nity. Thus lim t→+∞ r C (f , t) − t = for every f ∈ L p (Ω, F, P).
Theorem 6 yields that T has BH at every point f ∈ L p + .
Proof of Theorem 6. Let x ∈ C. In order to make the notation lighter, we write xn = T n x, n ≥ . We thus suppose that xn w − → . We will prove successively several equivalent formulations of the BH property for the sequence (xn), which will ultimately yield the result. Without loss of generality, we suppose that x = .
We claim that the following assertions are equivalent: 1. For every strictly increasing sequence (n k ) k≥ of integers, -The equivalence between ( ) and ( ) is essentially obvious: ( ) ⇒ ( ) is clear. In the converse direction, suppose that ( ) is not true: Then make the sets A k disjoint, and enumerate a suitable in nite subsequence of (A k ) as (n k ).
-The equivalence between ( ) and ( ) is not di cult either.
-Let us prove ( ) ⇒ ( ). Our assumption is that  xn k → , and we are done.
We can now conclude the proof of the theorem. We need the following fact. We have F(s + ) = lim d→+∞ x + z d ≤ r C (x, lim z d ). Now, lim z d = lim n∈B d xn ≤ F(s). Since the function t → r C (x, t) is increasing, F(s + ) ≤ r C (x, F(s)) and this proves Fact 8.
Using Fact 8, we have F(s + ) − F(s) ≤ r C (x, F(s)) − F(s). If F were increasing, we would be able to deduce that F(s + ) − F(s) tends to , and hence by the Cesàro theorem that Since F is not necessarily increasing, we replace F by the increasing functionF de ned byF(s) = max(F( ), . . . , F(s)), s ∈ N, and check that the same inequality as the one given in Fact 8 holds true for F. This concludes the proof of Theorem 6.
An improved version of Theorem 6, which is also perhaps more natural, can be of use in certain situations.
Theorem 9. [14] Let T ∈ B(X), with T ≤ and T(C) ⊂ C. Suppose that for every x ∈ C, Then T satis es the BH property at every x ∈ C.
Theorem 9 can be used to show that the space c = (u k ) k ∈ R N , lim k→+∞ u k exists , endowed with the norm . ∞, has the BH property ( [14]).

Spaces which do not have the BH property
Our aim is now to investigate spaces which do not have the BH property. We will present in particular a characterization, due to Lefèvre and Matheron [14], of the compact metric spaces K which are such that C(K) has the BH property.
Examples of spaces without the BH property: Proof. If K is a compact metric space, we denote by K the set of its accumulation points. This set K is nonempty as soon as K is in nite, and K is also a compact metric space.
The easy part of the proof is to show that if K is nite, let us say K = {a , . . . , a N }, then C(K) has BH. There exist disjoint compact sets K , . . . , K N such that K i = {a i } for every i = , . . . , N and K = ≤i≤N K i . Indeed, let V , . . . , Vn be open neighborhoods of a , . . . , a N respectively, such that their closures V , . . . , Vn are pairwise disjoint.
We have seen that c satis es the assumption of Theorem 9; it is easy to check that the direct ∞-sum of nitely many copies of c also satis es it, and hence has BH.
Conversely, let K be such that K is in nite. So K ≠ ∅. The simplest of these compact sets are the ones where K is reduced to one point. We study this case rst. Let S be a compact metric space such that where all the points s i,k are distinct, s i,k −→ i→+∞ s ∞,k for every k ∈ N and S k tends to s∞,∞ as k → +∞ in the sense that any neighborhood of s∞,∞ contains the sets S k for all but nitely many k's.
Indeed, if S = {s∞,∞}, S necessarily has this form: there exists (s ∞,k ) k such that S = s ∞,k ; k ∈ N , where s ∞,k → s∞,∞ as k → +∞. Let V k be a neighborhood of s ∞,k in S, with the sets V k , k ∈ N, disjoint, diam(V k ) < −k , and s∞,∞ ∉ V k . It is clear that the sets V k tend to s∞,∞ in the sense above. Let S k = V k ∩ S : S k = s ∞,k and hence there exists (s i,k ) i such that S k = s i,k ; i ∈ N ∪ s ∞,k , s i,k −→ i→+∞ s ∞,k . The set S\ k≥ S k {s∞,∞} is nite, so we add these few points to S , for instance, and we are done. Proposition 11. There exists a continuous map θ : S → S such that the contraction C θ : f → f • θ on C(S) does not have the BH property. Hence C(S) does not have the BH property.
Proof. De ne θ : S → S by setting Visually, the map θ acts as follows: Hence |A| n∈A C n θ f ∞ ↛ as |A| → +∞, and we have proved that C θ does not have the BH property.
The remaining part of the argument is rather general, and allows to pass from the space C(S) to any space C(K) with K in nite. It relies on several observations. Fact 12. If K is in nite, it contains a set S of the form above.
Proof. Since K is in nite, K ≠ ∅. Let s∞,∞ ∈ K . Let (V k ) k be a neighborhood basis of s∞,∞ with diam(V k ) < −k and V k \V k+ ≠ ∅ for every k. For each k, choose s ∞,k ∈ K ∩V k with s ∞,k ≠ s∞,∞. Without loss of generality, we can suppose that s ∞,k ∈ V k \V k+ . Let then (s i,k ) ⊂ K ∩ V k \V k+ (which is an open neighborhood of s ∞,k ) be such that s i,k → i s ∞,k with all the s i,k distinct and distinct from s ∞,k .
Finally, let S k := s i,k ; ≤ i ≤ +∞ : S k ⊂ V k and hence the sets S k tend to s∞,∞. The set S = s i,k ; ≤ i, k ≤ +∞ then satis es all the required properties.
Theorem 13 (Borsuk's linear isomorphic extension theorem). Let K be a compact metric space, and let E be a closed subset of K. There exists a bounded operator J : C(E) → C(K) such that J1 = 1, J = , and J(f ) |E = f for every f ∈ C(E).
In particular, J : C(E) → X := J(C(E)) is an isometry, and X is -complemented in C(K).
To see that X is -complemented in C(K), observe that P : C(K) → X de ned by Pg = J(g |E ), g ∈ C(K), is a projection of C(K) onto X.
Theorem 13 is a linear version of Tietze's extension theorem which states that for every f ∈ C(E), there exists g ∈ C(K) with g ∞ = f ∞ such that g |E = f . A proof of Theorem 13 can be found for instance in [6,Th. 4.4.4].
As a consequence of Fact 12 and Theorem 13, we obtain that C(S) is isometric to a -complemented subspace X of C(K).

Fact 14.
Let X be a -complemented subspace of a Banach space Z. If X fails the BH property, so does Z.
Proof. Let P : Z → X be a projection of Z onto X, with P = . Let T ∈ B(X) with T ≤ , and let x ∈ X be such that T n x w − → , but there exists a strictly increasing sequence (n k ) k of integers such that lim sup N→+∞ N N k= T n k x > .
Set T = T • P : Z → X ⊂ Z: then T ∈ B(Z), T ≤ , and T n z = T n (Pz) for every n ≥ and z ∈ Z. Hence T n x w − → . But the vector x ∈ X ⊂ Z satis es Px = x and lim sup N→+∞ N N k= T n k x > .
By Proposition 11, C(S) fails BH, and hence C(K) fails BH as well. This concludes the proof of Theorem 10.