A new way of constructing examples in operator ergodic theory

We propose a method of constructing examples in operator ergodic theory which unifies and extends some previously known examples. It also allows us to answer several questions that have been open for some time (including a question of Allan [1]).


Introduction
Let T be a bounded linear operator on a complex Banach space X. We say that T is Cesàro ergodic if A n (T ) 1 n + 1 n k=0 T k converge strongly on X. (1·1) Then X = Ker(T − I) ⊕ Im(T − I), and the limit operator P is the projection of X onto Ker(T − I) along Im(T − I); see e.g. [13,corollary VIII·5·2]. It is well known that T is Cesàro ergodic if it is power bounded, that is, and the space X is reflexive. However, (1·2) is not necessary for (1·1) even if X is a Hilbert space [11, p. 453, remark 1]; see also Example 3·1 below. On the other hand, the Cesàro boundedness of T, that is as well as the condition lim n→∞ 1 n T n x = 0, for every x ∈ X, are certainly necessary for (1·1). We note that (1·3) and (1·4) are, in general, mutually independent; see e.g. [17, remark 4]. If the space X is reflexive, then, conversely, (1·3) together with (1·4) imply (1·1); see [13,corollary VIII·5·4]. If only (1·3) is assumed, then (1·1) may fail as the well-known Assani example, shows [15]. It is not clear to what extent the Cesàro bounded operators share ergodic properties with the power bounded operators. The subtlety of the question can be illustrated by the fact that a certain 'quasi-ergodic' theorem always holds for an arbitrary uniformly bounded family of operators on a Hilbert space, see [9, lemma 4].
On the positive side,Émilion proved in [15] that (1·1) is true for any positive Cesàro bounded operator on a reflexive Banach lattice (in particular on l 2 ). Here the reflexivity cannot be dropped in general [18]. This result was generalized by Derriennic in [11], where the positivity of T was replaced by a certain geometrical condition on {T n : n 0}. However, in the same paper, Derriennic showed that Cesàro bounded operators (even on Hilbert spaces) may have rather 'pathological' behaviour of Cesàro averages in contrast to the well-studied power bounded operators. We present these examples in a unified and simplified way, strengthen some of them and find new ones. Suppose T is Cesàro ergodic. The restriction of T to Ker(T − I) is I, and so has a very simple form. The restriction T 0 of T to Im(T − I) satisfies lim n→∞ A n (T 0 )x = 0, for every x ∈ Im(T − I). In general, nothing more can be said about T 0 . However, under additional spectral assumptions, quite strong conclusions can be made. In 1986, Katznelson and Tzafriri [22] proved that if T is a power bounded linear operator on a Banach space X, and , this was proved earlier by Esterle in [16]. Actually, the general case can be reduced to this one as was noticed by Vũ [34].) Thus, under (1·2) and (1·6), we have T 0 n → 0, n → ∞ strongly on Im(T − I), and if X is reflexive, then the powers of T converge strongly on X, a conclusion stronger than (1·1). The Esterle-Katznelson-Tzafriri theorem substantially influenced not only abstract ergodic theory (e.g. uniform ergodic theorems), but also operator theory (e.g. asymptotics of operator semigroups) and function theory (e.g. Tauberian theorems).
It is natural to want to understand the limits of such a good result. With this in mind, note that (1·7) implies (1·6) in view of the spectral mapping theorem, and it also implies In 1989, Allan asked whether a bounded linear operator T on a Banach space X satisfying (1·8) and the "extremal" spectral condition, also has property (1·7); see [1, p. 7] (the formulation contains a misprint but the right formulation has been communicated by Allan orally). We show that the answer to this question is negative. Moreover, our technique allows us to construct a Banach space X and a bounded linear operator T on X such that the properties (1·3), (1·6) and (1·8) hold, but for every m ∈ N {0}. Thus, the Esterle-Katznelson-Tzafriri theorem fails in a dramatic way if the condition of power boundedness of T is replaced by the nearby conditions (1·3) and (1·8). We denote by L(X) the space of bounded linear operators on X, and by σ(T ) and R(λ, T ) (T − λI) −1 the spectrum and the resolvent of T, respectively. The identity operator on different Banach spaces will always be denoted by the same symbol I.

A general construction
All examples in this paper will depend on the following transparent and simple matrix construction. In some sense, it is an 'inverse' of a construction proposed in [8]. It emerges naturally if one tries to understand the different approaches to the Ritt condition (the property (2·2) below) studied recently in [7,14,[24][25][26]28]. Note also [3, section 3] where the ergodicity of matrix semigroups was studied.
Consider the Banach space X = X ⊕ X with the norm Note that if X is a Hilbert space, then X is a Hilbert space too. Let the bounded linear operator T on X be defined by the operator matrix where T ∈ L(X). The operator T is an extension of T to the larger space X ⊃ X and coincides with T on its invariant subspace X.
Lemma 2·1. We have: it suffices to observe that lim n→∞ T n − T n+1 = 0 implies lim n→∞ (2·4) If A n (T ) is bounded, then from (2·4) we first get that A n (T ) is bounded, and then, looking at the right upper corner of that matrix, we conclude that sup n 0 T n < ∞. The converse is also clear from (2·4).
(vii) For the proof of (vii), observe that Then the latter matrix representation of T n (T − I) m gives the conclusion.
(viii) This follows directly from (i) and the matrix representation The gap between Cesàro bounded and Cesàro ergodic operators T in this construction is the gap between operators T with uniformly bounded powers and with strongly convergent powers. The latter one is quite large.
Remark 2·2. All the statements of the above lemma remain true if one replaces T by T * , and, correspondingly, T by and X by X * = X * ⊕ X * . The verification is left to the reader.
Remark 2·3. Observe that if there exists T −1 ∈ L(X), then A reasoning similar to the proof of Lemma 2·1(iii) shows that T −1 is Cesàro bounded if and only if T −1 is power bounded.
As an immediate illustration of Lemma 2·1 we give the following statement.
Corollary 2·5. Let K be a compact subset of the unit disc {λ ∈ C : |λ| 1} such that the intersection K {λ ∈ C : |λ| = 1} is nonempty. Then there exist a Hilbert space X and a Cesàro bounded but not power bounded linear operator T on X such that where {λ n : n 1} = K. Note that T 1 is a contraction on X 1 , and σ(T 1 ) = K. Since the intersection σ(T 1 ) {λ ∈ C : |λ| = 1} is nonempty by assumption, it must contain a point different from 1, consequently, the operator T 1 does not satisfy the Ritt condition. Thus the corresponding matrix operator T T 1 on the Hilbert space X X 1 is not power bounded by Lemma 2·1(vi), but it is Cesàro bounded in view of Lemma 2·1(iii). Since σ(T 1 ) = K, we have σ(T ) = K by Lemma 2·1(i). Now, let 1 ∈ K. Consider the Volterra operator (Alternatively, one can apply [25, theorem, p. 137].) By Lemma 2·1(i), we have σ(T 2 ) = {1}. By Lemma 2·1(iii),(vi), the matrix operator T 2 corresponding to T 2 is Cesàro bounded but not power bounded. Then the operator T T 1 ⊕ T 2 on the Hilbert space X X 1 ⊕ X 2 is Cesàro bounded but not power bounded. Moreover, Note that the assumptions of Corollary 2·5 are necessary for the conclusion. For Hilbert space nonunitary contractions the spectral conclusion of Corollary 2·5 is essentially well known. See, for example, [30, theorem 2] for the proof.
The following simple estimate for the norms of operator matrices will be used frequently.
The proof is straightforward and is, therefore, omitted.

Examples
In this section we will present a series of examples showing a striking difference between 'ergodic' behaviour of power bounded and Cesàro bounded operators.
Define the Hilbert space X as the orthogonal sum of two copies of l 2 , that is X l 2 ⊕ l 2 , where l 2 will mean either l 2 (N, C) or l 2 (Z, C).
According to the mean ergodic theorem [23, theorem 2·1·1], an operator is Cesàro ergodic if it is power bounded and its Cesàro averages are weakly convergent. On a reflexive space, a power bounded operator and its adjoint are simultaneously Cesàro ergodic (see the Introduction). Examples 3·1 and 3·3 below show that these two properties may fail if the operator in question is merely Cesàro bounded, even on a Hilbert space.
Example 3·1. There exists a T ∈ L(X ) such that: (a) A n (T ) converge strongly; (b) A n (T * ) do not converge strongly; (c) T n 2n for all n ∈ N. Let X = l 2 (N, C). Let T ∈ L(X) be the backward shift: , the corresponding matrix operator T on X satisfies (a). However, T * n x do not converge in X for all x ∈ X. Thus T * satisfies (b) by Remark 2·2 and Lemma 2·1(iv). Lemma 2·6 and (2·3) yield hence T satisfies also (c).
Moreover, we can conclude that A n (T * )x do not converge for any x of the form x = x ⊕ x ∈ X , x0. Indeed, observe that for such x we have which is divergent.
Remark 3·2. Note that T n 2n, n ∈ N, is clearly the fastest possible growth for a Cesàro bounded operator.
Example 3·3. There exists a T ∈ L(X ) such that: (a) both A n (T ) and A n (T * ) converge weakly; (b) neither A n (T ), nor A n (T * ) converge strongly; (c) T n 2n for all n ∈ N. Let X be l 2 (Z, C). Let T ∈ L(X) be the backward shift: Let T be the operator corresponding to T by Lemma 2·1. By applying Lemma 2·1(iv),(v) to the contractions T, T * , respectively, we conclude that T has the properties (a) and (b). As before, (2·3) yields T n 2n, n 1.
Moreover, A n (T * )x diverge for any nonzero x ∈ X of the form x = x ⊕ x.
Remark 3·4. Operators possessing property (b) of the above example can be constructed already on the two-dimensional space C 2 using the Assani matrix (1·5). However, Lemma 2·1 makes it possible to provide such examples with various additional properties. We shall not go into details here.
The next example concerns mixing properties of Cesàro bounded operators. If T is a power bounded operator on a Banach space, then the averages n −1 n i=1 T ki are uniformly bounded for all strictly increasing subsequences {k i : i 1} of N {0}. This makes it possible to study ergodic properties of T along subsequences (see, for example, [5, p. 242] and the references therein). However, if T is merely Cesàro bounded, then the averages n −1 n i=1 T ki may be unbounded along some subsequence Example 3·5. There exists a T ∈ L(X ) such that: (a) A n (T ) converge strongly; To see this it suffices to consider T with T being the backward shift on on l 2 (N, C). Consider The classical Sz.-Nagy similarity criterion states that if a bounded linear operator T on a Hilbert space satisfies sup n∈Z T n < ∞, then T is similar to a unitary operator [33]. So ergodic properties of T in this case are essentially the same as of the corresponding unitary operator. We show in the next example that the version of the Sz.-Nagy criterion for Cesàro bounded operators is not true. This is in contrast to the Gelfand-Hille theorems where the assumption on the growth of the powers can be replaced by the same assumption on the Cesàro means [12].
Remark 3·8. Another counterexample to the version of the Sz.-Nagy similarity criterion for Cesàro bounded operators was obtained in [10, p. 64]. This counterexample assumes Cesàro boundedness even in a stronger sense than our Example 3·7. However, the growth of powers of the operator considered there cannot be made faster than √ n, while in the above example the growth of powers is extremal.
It would be interesting to find T (not necessarily of the form (2·1)) such that in Examples 3·1 and 3·3 the divergence takes place at all nonzero elements of X . Let T (k) be a multiplication operator on L 2 ((0, 1)) defined by
Remark 4·2. The construction of Example 4·1 shows that, moreover, the operator T is Cesàro ergodic. Indeed, observe that for every k ∈ N the powers T (k) n converge to zero strongly on L 2 ((0, 1)). Therefore, by the reasoning in Lemma 2·1(iv), the averages A n (T (k)) converge to zero strongly on X . Then A n (T)x converge to zero for all x ∈ X with finite support. Since the set of such x is dense in X and (4·12) holds, we conclude that A n (T) converge to zero on the whole of X.
The next example answers in the negative a question of Allan from [1, p. 7]. Its construction relies on special properties of the Volterra operator and was influenced by [19]. First we shall need several asymptotic properties of the Laguerre polynomials with parameter k ∈ N {0}; see [32, pp. 177 and 198].
Lemma 4·4. Let k ∈ N {0}. Then: We shall also need the following formula for the norm of an integral operator on L 1 ((0, 1)). It follows from a very general result given in [21,theorem XI·1·4]. To make the formula more accessible we provide it with a particularly simple proof.
Example 4·6. There exists a Banach space X and a bounded linear operator T on X such that: As in the proof of Corollary 2·5 again take the Volterra operator this time on X = L 1 ((0, 1)).
Further, we apply Example 4·6 to the study of Cesàro means of bounded operators with one point spectrum. The next example answers a question from [12, remark 6]. It is an iteration of the construction (2·1).

Open problems and final remarks
(1) It is not clear whether or not one can construct a Banach space X and a Cesàro bounded linear operator T on it such that σ(T ) = {1}, but at the same time T n (T − I) −→ 0, n → ∞; see also [35, p. 378]. If there are no such X and T, then using Lemma 2·1, we can conclude that any power bounded operator T on a Banach space X with σ(T ) = {1} has the property to coordinate-wise order. Moreover, T is positive on X if and only if T I on X. If, in addition, T is Cesàro bounded, then T = I by Lemma 2·1(iii) and [6, theorem 1]. The consideration of Cesàro unbounded operators T excludes any (Cesàro) mean ergodic theorems for T . One may replace the element T − I of (2·1) by the element I − T.
Then the corresponding matrix operator T is positive on X whenever O T I. If X has order continuous norm (e.g. if X = L p (X, dµ), 1 p < ∞), then T n converge strongly on X; see [31, pp. 89-92]. Similarly to Lemma 2·1(iv), the operator T is Cesàro ergodic on X .
(5) It would be interesting to know whether the famous Jacobs-de Leeuw-Glicksberg decomposition theorem is true for Cesàro bounded operators. Recall one of its versions which can be found in [20, theorem 9]; see [23, theorem 2·4·4] for the general case.
Could this statement be true if, instead of T 1, we assume that A n (T ) 1, n 0? (6) Suppose that every power bounded operator T ∈ L(X) is Cesàro ergodic. It is still unknown whether X must be reflexive. Suppose now that even every T ∈ L(X) from the a priori larger class of operators satisfying (1·3) and (1·4) is Cesàro ergodic. Does it follow that X is reflexive? (7) Since the mid-1990s. Atzmon has been claiming the existence of another example (an essentially different approach) answering negatively Allan's question; see [35, p. 373]. However, the details have not been available until now.