Maximum Modulus Principle For Multipliers and Mean Ergodic Multiplication Operators
aa r X i v : . [ m a t h . F A ] A ug Maximum Modulus Principle For Multipliers andMean Ergodic Multiplication Operators
Eugene Bilokopytov ∗ August 11, 2020
Abstract
The main goal of this note is to show that (not necessarily holomorphic) mul-tipliers of a wide class of normed spaces of continuous functions over a connectedHausdorff topological space cannot attain their multiplier norms, unless they areconstants. As an application, a contractive multiplication operator is either a mul-tiplication with a constant, or is completely non-unitary. Additionally, we explorepossibilities for a multiplication operator to be (weakly) compact and (uniformly)mean ergodic.
Keywords:
Function Spaces, Multiplication Operators, Mean Ergodic Opera-tors;MSC2020 46B20, 47A35, 47B38
Normed spaces of functions are ubiquitous in mathematics, especially in analysis. Thesespaces can be of a various nature and exhibit different types of behavior, and in this paperwe discuss some questions related to these spaces from a general, axiomatic viewpoint.One of the most prominent classes of linear operators on the spaces of functions is theclass of multiplication operators (MO). In this article we continue our investigation (see[2, 3]) of the general framework which allows to consider any Banach space that consistsof continuous (scalar-valued) functions, such that the point evaluations are continuouslinear functionals, and of MO’s on these spaces.First, let us define precisely what we mean by a normed space of continuous functions.Let X be a topological space (a phase space ) and let C ( X ) denote the space of allcontinuous complex-valued functions over X endowed with the compact-open topology.A normed space of continuous functions (NSCF) over X is a linear subspace F ⊂ C ( X )equipped with a norm that induces a topology, which is stronger than the compact-opentopology, i.e. the inclusion operator J F : F → C ( X ) is continuous, or equivalently the ∗ Email address [email protected], [email protected]. B F is bounded in C ( X ). If F is a linear subspace of C ( X ), then the pointevaluation at x ∈ X on F is the linear functional x F : F → C , defined by x F ( f ) = f ( x ).If F is a NSCF, then all point evaluations are bounded on F . Conversely, if F ⊂ C ( X ) isequipped with a complete norm such that x F ∈ F ∗ , for every x ∈ X , then F is a NSCF.We will call a NSCF F over X (weakly) compactly embedded if J F is a (weakly) compactoperator, or equivalently, if B F is (weakly) relatively compact in C ( X ). Clearly, everycompactly embedded NSCF’s is weakly compactly embedded. On the other hand, anyreflexive NSCF is also weakly compactly embedded. By a Banach space of continuousfunctions (BSCF) we mean a complete NSCF.A multiplication operator (MO) with symbol ω : X → C is a linear map M ω on thespace F ( X ) of all complex-valued functions on X defined by[ M ω f ] ( x ) = ω ( x ) f ( x ) , for x ∈ X . Let F and E be NSCF’s over X . If M ω F ⊂ E , then we say that M ω isa multiplication operator from F into E (we use the same notation M ω for what is infact M ω | F ). If in this case F = E , then we will call ω a multiplier of F . If both F and E are BSCF’s, then any MO between these spaces is automatically continuous due toClosed Graph theorem. However, in concrete cases it can be very difficult to determineall MO’s between a given pair of NSCF’s, and in particular to characterize all multipliersof a NSCF (see e.g. [13] and [19], where the multiplier algebras of some specific familiesof NSCF’s are described).In this article we explore how certain properties of MO’s are reflected on their symbols.In [3] we focused on MO’s which are surjective isometries and isometries in general. Inparticular, we proved that on a wide class of NSCF’s every isometric MO with a non-constant symbol is completely non-unitary, i.e. has no unitary restrictions. The methodsemployed there were of Banach space geometry. In the present article we attack thesame problem using a topological approach. This allows to extend the mentioned resultto contractions, and also somewhat enlarge the class of admissible NSCF’s (see Corollary2.6). In the process we discover a result reminiscent of the Maximum Modulus Principlefrom the Complex Analysis (see theorems 2.4 and 2.5).In addition to contractive MO’s we also consider the MO’s which are (weakly) compactand (uniformly) mean ergodic. In particular, we show that a MO can be compact only indegenerate cases (see Proposition 2.10) and show that a NSCF that has “a lot” of weaklycompact MO’s has to be weakly compactly embedded itself (Proposition 2.8). We alsoabstract some of the results from [4] and [5] about mean ergodic MO’s (see Theorem 3.2,Proposition 3.7 and Corollary 3.8). This involves proving one more version of “MaximumModulus Principle” for multipliers. We conclude with an alternative proof of the resultfrom [4] saying that a MO on a weighted Banach space of continuous functions is meanergodic if and only if it is uniformly mean ergodic.Throughout the paper by Id X we mean the identity map on a set X , while thesupremum norm of f : X → C is denoted by k f k ∞ .2efore concluding this section with some concrete examples of NSCF’s, let us mentiona large class of compactly embedded NSCF’s. If X is a domain in C n , i.e. an openconnected set, and F is a NSCF over X that consists of holomorphic functions, then F is compactly embedded. Indeed, by Montel’s theorem (see [18, Theorem 1.4.31]), B F is relatively compact in C ( X ), since it is a bounded set that consists of holomorphicfunctions. Example . The Hardy space H is the BSCF over the (open) unit disk D ⊂ C thatconsists of holomorphic functions f with the norm defined by k f k = ∞ P n =0 | a n | , where { a n } ∞ n =0 are the Taylor coefficients of f . One can show that this is a Hilbert space withmonomials forming an orthonormal basis. The Hardy space is among the most studiedfunction spaces, and we refer to e.g. [15] for more information. Example . Assume that X is a Tychonoff space and let u : X → (0 , + ∞ ) be uppersemi-continuous. Define the weighted space of continuous functions C ∞ u = { f ∈ C ( X ) , k f k ∞ u = k uf k ∞ < + ∞} . One can show that this is a BSCF over X with respect to the norm k · k ∞ u . Moreover, k x C ∞ u k = u ( x ) , for every x ∈ X . Indeed,it is immediate that | f ( x ) | ≤ u ( x ) , for every f ∈ B C ∞ u , while since u is lower semi-continuous and positive-valued, for any ε > f : X → (0 , + ∞ )such that f ≤ u , while f ( x ) ≥ u ( x ) − ε (see [6, IX.1.6, Proposition 5]). One can alsoshow that C ∞ u is not weakly compactly embedded (the proof from [Example 2.6][3] carriesover to the case when u is non-constant). In the case u ≡ C ∞ ( X ).Additionally, C u consists of f ∈ C ∞ u such | uf | vanish at infinity. Example . Assume that X is a domain in C n and let u : X → (0 , + ∞ ) be uppersemi-continuous. Define the weighted space of holomorphic functions H ∞ u = C ∞ u ∩ H ( X ),where H ( X ) is the subspace of C ( X ) that consists of holomorphic functions. As wasmentioned before, this space is compactly embedded, and is a closed subspace of C ∞ u .Note however, that the equality k x C ∞ u k = u ( x ) may not hold (see [1]). In the case u ≡ H ∞ ( X ). Additionally, H u = H ∞ u ∩ C u . Example . Assume that X is a bounded domain in C n . Then A ( X ) is the closedsubalgebra of H ∞ ( X ) which consists of functions that admit a continuous extension on X . Another natural way to represent this space is A (cid:0) X (cid:1) which is the closed subalgebraof C ∞ (cid:0) X (cid:1) that consists of functions holomorphic on X . Then A ( X ) is a BSCF over X , and A (cid:0) X (cid:1) is a BSCF over X . Both contain constant functions. Note that therecan be no ambiguity in our notations since X is open and X is compact. Also, notethat MO’s are same on these BSCF’s, despite being induced by functions on X and X respectively. In this section we consider some basic properties of MO’s, including the Maximal ModulusPrinciple for multipliers. Everywhere in this section X is a Hausdorff space. Note that3 ω is continuous on F ( X ), for every ω : X → C , and is continuous on C ( X ) whenever ω is continuous. We will however mostly deal with MO’s on NSCF’s. These can becharacterized by the following well-known fact (see e.g. [2, Proposition 2.4 and Corollary2.5]). Proposition 2.1.
Let F and E be NSCF’s over X and let T ∈ L ( F , E ) . Then T = M ω ,for ω : X → C , if and only if T ∗ x E = ω ( x ) x F , for every x ∈ X . In other words, T is aMO if and only if T ∗ x E ⊂ C x F , for every x ∈ X . If in particular F = E , then T is a MO if and only if x F is an eigenvector of T ∗ (orelse x F = 0 F ∗ ), for every x ∈ X . Then the multiplier is the correspondence between x and the eigenvalue of T ∗ for x F .For Y ⊂ X define F Y = { f ∈ F | supp f ⊂ Y } = { x F , x ∈ X \ Y } ⊥ , which is a closedsubspace of F , and is also a NSCF over X . It follows that Ker M ω = F ω − (0) , M ω F ⊂ F X \ ω − (0) and M ω F Y ⊂ F Y , for any Y ⊂ X .Note that in general we cannot reconstruct the symbol of a MO from its data as alinear operator between certain NSCF’s, in the sense that the equality of MO’s does notimply the equality of their symbols. In order to prevent this pathology from happeningwe have to introduce the following concept. We will call a NSCF F over X independent if for every x ∈ X we have x F = 0 F ∗ , i.e. there is f ∈ F such that f ( x ) = 0. It is easyto see that a MO from a 1-independent NSCF determines its symbol. Moreover, someproperties of this symbol can also be recovered. Proposition 2.2. If F is a -independent NSCF over X , then its multipliers are con-tinuous. Moreover, k ω k ∞ ≤ k M ω k , for every multiplier ω of F .Proof. Let x ∈ X . Since F is 1-independent, there is f ∈ F such that f ( x ) = 0. As f and M ω f are continuous, ω is continuous at x as a ratio of continuous functions. Moreover,if M ω ∈ L ( F ), then M ∗ ω x F = ω ( x ) x F implies that | ω ( x ) | ≤ k M ∗ ω k = k M ω k . Since x waschosen arbitrarily, the result follows.Let us now proceed to the main result of the paper. We need the following lemma. Lemma 2.3.
Let ω : X → C be such that for every x ∈ ω − (1) there is f ∈ C ( X ) with f ( x ) = 0 and ωf = f . Then ω − (1) is open.Proof. Let x ∈ ω − (1) and let f ∈ C ( X ) be such that f ( x ) = 0 and ωf = f . The lattermeans that f ( y ) ( ω ( y ) −
1) = 0, for every y ∈ X . Since f ( x ) = 0 there is an open subset U of X containing x such that f ( y ) = 0, for any y ∈ U . Hence, ω ( y ) − y ∈ U , from where U ⊂ ω − (1), and so x ∈ int ω − (1). Since x was chosen arbitrarily,we conclude that ω − (1) is open.Recall that a subset Y ⊂ X is called clopen if it simultaneously closed and open. Let T = ∂ D be the unit circle. 4 heorem 2.4. Let F be a -independent NSCF over X . If ω : X → C is such that M ω isan operator of norm on F , and moreover M ω B F is relatively weakly compact in C ( X ) ,then ω − ( λ ) is clopen, for every λ ∈ T . Furthermore, if additionally X is connected andthere is x ∈ X such that | ω ( x ) | = 1 , then ω ≡ λ , for some λ ∈ T .Proof. Let λ ∈ T . Replacing ω with λω if needed we may assume that λ = 1. Since ω iscontinuous due to Proposition 2.2, ω − (1) is closed. Let x ∈ ω − (1).Since M ω B F is relatively weakly compact in C ( X ), it follows (see [12, 4.3, Corollary2]) that B = M ω B F C ( X ) is a convex pointwise compact set in C ( X ). Since k M ω k ≤ M ω B F ⊂ B F , from where M ω M ω B F ⊂ M ω B F , and since M ω is continuous on C ( X ) we get M ω B ⊂ B . Due to pointwise compactness of B , the functional x F attains itsmaximum on it, which is equal to sup f ∈ B F | [ M ω f ] ( x ) | = k M ∗ ω x F k = k ω ( x ) x F k = k x F k > D = { f ∈ B | f ( x ) = k x F k } , which is thus a non-empty convex pointwisecompact set in C ( X ). For every f ∈ D we have [ M ω f ] ( x ) = ω ( x ) f ( x ) = k x F k . Since M ω B ⊂ B , this implies that M ω D ⊂ D . Therefore, M ω | D is a pointwise continuous self-map of a pointwise compact convex set D . Hence, by Tychonoff Fixed Point Theorem(see [11, Theorem 12.19]), there is f ∈ D such that M ω f = f , i.e. ωf = f .Since x ∈ ω − (1) was chosen arbitrarily, from Lemma 2.3, ω − (1) is open, andtherefore clopen.If X is connected, every clopen set is either empty, or equal to X . Hence, if ω ( x ) = λ ,for some λ ∈ T , the set ω − ( λ ) is clopen and nonempty, and so ω ≡ λ .Let J F be the embedding of F into C ( X ). The condition of relative weak compactnessof M ω B F in C ( X ) is equivalent to weak compactness of J F M ω as an operator from F into C ( X ). This operator is weakly compact whenever either J F or M ω is. In particular,we get the following “Maximum Modulus Principle for Multipliers”. Theorem 2.5.
Let X be connected and let F be a -independent NSCF over X . Let ω : X → C be a non-constant multiplier of F . Assume furthermore that either F is weaklycompactly embedded, or M ω is a weakly compact operator on F . Then | ω ( x ) | < k M ω k ,for every x ∈ X . Corollary 2.6. [cf. [3, Theorem 3.17] ] Let X be connected and let F be a -independentweakly compactly embedded NSCF over X . If ω : X → C is a non-constant function suchthat k M ω k ≤ on F , then it is completely non-unitary, i.e. has no unitary restrictionsto proper subspaces.Proof. Let E be a non-trivial subspace of F (and so a NSCF) such that the restriction of M ω on E is unitary. Let x be such that x E = 0 E ∗ . Then ( M ω | E ) ∗ is an isometry, and so0 = k x E k = k ( M ω | E ) ∗ x E k = k ω ( x ) x E k = | ω ( x ) | k x E k , from where | ω ( x ) | = 1 = k M ω k .Contradiction.As an application we can conclude that non-trivial isometric multiplication operatorshave a Wold decomposition (see [17]) into a direct sum of shifts.5 xample . The classic example of a MO acting like a shift on a space of functions isthe multiplication with a free constant on the Hardy space. Namely, let H be the Hardyspace over D , and let ω = Id D . Then it is easy to see that M ω is an isometry from H ontothe 1-co-dimensional subspace H D \{ } . Also, one can show that M ω is an isometry on H if ω is an infinite Blashke product (see [15, IV.A]). In this case M ω H ⊂ H D \ ω − (0) , whichis of infinite co-dimension, since ω − (0) is infinite, and point evaluations are linearlyindependent on H . Therefore, M ω is an infinite sum of unit shifts.As was mentioned above, the condition of relative weak compactness of M ω B F isalways satisfied if F is weakly compactly embedded. It turns out that this implicationcan be reversed in some sense. Proposition 2.8.
A NSCF F over X is weakly compactly embedded if and only if forevery x ∈ X there is a multiplier ω of F such that ω ( x ) = 0 and M ω B F is relativelyweakly compact in C ( X ) .Proof. Necessity: If F is weakly compactly embedded, ω ≡ F thatdoes not vanish and such that M ω B F = B F is relatively weakly compact in C ( X ).Sufficiency: Let J be the embedding of F into F ( X ). Let x ∈ X , and let ω be amultiplier of F such that ω ( x ) = 0 and B = M ω B F is relatively weakly compact in C ( X ).Let M be the multiplication operator with symbol ω defined on F ( X ) (we keep labelingthe corresponding operator on F by M ω ).Since F ( X ) is reflexive we have M ∗∗ = M . Moreover, if T = J M ω = M J , then
M J ∗∗ B F ∗∗ = T ∗∗ B F ∗∗ = T B F (the proof of [10, VI.4, Theorem 2] carries over to the casewhen the target space is locally convex). The latter set is in fact equal B F ( X ) , and since B is weakly compact in C ( X ), it follows that B C ( X ) is pointwise compact, from where M J ∗∗ B F ∗∗ = B F ( X ) = B C ( X ) ⊂ C ( X ).Hence, if f ∈ B F ∗∗ , then g = J ∗∗ f is a function on X such that ωg is continuous.Since ω is continuous and ω ( x ) = 0, g is continuous at x . Since x and f were chosenarbitrarily, it follows that J ∗∗ B F ∗∗ = B F F ( X ) ⊂ C ( X ), and so B F is relatively pointwisecompact in C ( X ) and so is relatively weakly compact in (see [12, 4.3, Corollary 2]).Note that both of the conditions in the proposition depend on the way F sits in C ( X ).It is natural to wonder if one can obtain an equivalence between stronger conditionsintrinsic for F : Question 2.9. If X is connected, is it true that a BSCF F over X is reflexive if andonly if for every x ∈ X there is ω : X → C such that ω ( x ) = 0 and M ω is a weaklycompact operator on F ? Of course, necessity is obvious, since the Id F is weakly compact as soon as F isreflexive. Without the assumption that X is connected the answer is negative: consider F = l as a non-reflexive NSCF over N , and then ω : N → C defined by ω ( n ) = n doesnot vanish and generates a compact multiplication operator. Strong evidences in favourof the affirmative answer to the question are Corollary 2.11 below and Remark 3.10.6hile being weakly compact is a rather common property of multiplication operators(for example this is the case for all MO’s if F is reflexive), compactness occurs only intrivial cases. Proposition 2.10.
Let F be a -independent NSCF over X . If ω : X → C is such that M ω is a compact operator on F , then ω ( X ) is either finite, or is a sequence of numbersthat converges to . Moreover, ω − ( λ ) is clopen in X , for every λ ∈ C \ { } , and ω isconstant on every component of X . Furthermore:(i) If X is connected and dim F = ∞ , then ω ≡ .(ii) If the set of point evaluations on F is linearly independent, then X \ ω − (0) is at mostcountable collection of isolated points in X .Proof. Recall that ω ( x ) is an eigenvalue of M ∗ ω , for every x ∈ X . If M ω is compact,then so is M ∗ ω (see [11, Theorem 15.3]), from where ω ( X ) is contained in a sequence ofcomplex numbers that converges to 0 (see [11, Corollary 15.24]). Hence, { λ } is clopen in ω ( X ), for every λ ∈ ω ( X ) \ { } . Since ω is continuous, it follows that ω − ( λ ) is clopenin X .If Y is a component of X , and λ ∈ ω ( X ) \ { } , then ω − ( λ ) ∩ Y is clopen in Y . Sincethe latter is connected, it follows that either ω | Y ≡ λ , or ω − ( λ ) ∩ Y = ∅ . Consequently,either ω | Y ≡
0, or ω − (0) ∩ Y = ∅ . Hence, ω is constant on every component of X .(i): If X is connected, ω is a constant function, and so M ω = λId F , for some λ ∈ C .In the case when dim F = ∞ the only value of λ compatible with compactness of λId F is 0.(ii): Let λ ∈ ω ( X ) \ { } . Due to compactness, the eigenspace of M ∗ ω correspondingto λ is finitely dimensional (see [11, Corollary 15.24]). Since the point evaluations at theelements of ω − ( λ ) belong to that eigenspace and are linearly independent, it followsthat ω − ( λ ) is a clopen finite set, and so consists of isolated points.In the special case when F has the Dunford-Pettis property every weakly compactoperator on F is completely continuous, and so every square of a weakly compact operatoris compact. Observe that M ω = M ω , and the conclusions of Proposition 2.10 hold for ω if and only if they hold for ω . Therefore, we get the following result. Corollary 2.11.
The conclusions of Proposition 2.10 hold if F has the Dunford-Pettisproperty and M ω is merely weakly compact. In particular, it follows that there is no non-zero weakly compact multiplication oper-ators on many classical function spaces that satisfy Dunford-Pettis condition, includingweighted spaces of continuous functions, some of the weighted spaces of holomorphicfunctions, the ball and (poly)disk algebras, some of the Sobolev spaces, weighted lit-tle Lipschitz spaces, the (little) Bloch space and the Bergman and Besov spaces withexponent 1.While the Hardy space H does not have the Dunford-Pettis property, it is knownthat every weakly compact weighted composition operator on it is automatically compact,from where we again can see that there are no non-zero weakly compact multiplicationoperators on H . The same property also holds for many of the Lipschitz spaces.7 Mean Ergodic Multiplication Operators
In this section we will discuss possibilities for multiplication operators to be mean er-godic or uniformly mean ergodic. Our proofs are mostly inspired by the methods in[4] and [5], where similar questions were investigated for MO’s on weighted spaces ofcontinuous and holomorphic functions. Recall that a continuous linear operator T ona Banach space E is called (resp uniformly ) mean ergodic if there is P ∈ L ( E ) suchthat n n P k =1 T k f n →∞ −−−→ P f , for every f ∈ E (resp n n P T n n →∞ −−−→ P in L ( E )). Also, recallthat T ∈ L ( E ) is called power-bounded if sup n ∈ N k T n k < + ∞ , and more generally Cesarobounded if sup n ∈ N (cid:13)(cid:13)(cid:13)(cid:13) n n P k =1 T k (cid:13)(cid:13)(cid:13)(cid:13) < + ∞ . Let us also recall a characterization of mean ergodicity(see e.g. [16, Chaper 2, theorems 1.1, 1.3, 1.4 and 2.1]). Theorem 3.1 (Mean Ergodic Theorem) . Let T be a power-bounded operator on a Banachspace E . Then the following are equivalent:(i) T is mean ergodic;(ii) The sequence (cid:26) n n P k =1 T k f (cid:27) n ∈ N has a weak cluster point, for every f ∈ E ;(iii) For every ν ∈ E ∗ \ { E ∗ } such that T ∗ ν = ν there is f ∈ E such that T f = f and h f, ν i 6 = 0 ;(iv) E ∼ = Ker ( Id E − T ) ⊕ ( Id E − T ) E .Furthermore, T is uniformly mean ergodic if and only if it is mean ergodic and ( Id E − T ) E is closed. Throughout this section F is a 1-independent BSCF over a Hausdorff space X . For ω : X → C define ω n = n n P k =1 ω k . Note that ω n ( x ) = 1 if ω ( x ) = 1, and ω n ( x ) = ω ( x )(1 − ω ( x ) n ) n (1 − ω ( x )) otherwise. If k ω k ∞ ≤ ω n n →∞ −−−→ ω − (1) pointwise; if | ω ( x ) | >
1, then | ω n ( x ) | n →∞ −−−→ + ∞ . It follows immediately from identities M nω = M ω n and n n P k =1 M kω = M ω n and Proposition 2.2 that if M ω is a Cesaro bounded operator on a 1-independentNSCF F over X , then k ω k ∞ ≤ ω is a multiplier of F , then M ω | F ω − = Id F ω − , and M ω F Y ⊂ F Y , for any Y ⊂ X . Theorem 3.2. If F is -independent and ω : X → C then:(i) If M ω is a mean ergodic operator on F , then k ω k ∞ ≤ , ω − (1) is clopen, F ∼ = F ω − (1) ⊕ F X \ ω − (1) and M − ω | F X \ ω − is injective and has a dense range.(ii) If M ω is uniformly mean ergodic, then additionally inf x ∈ X \ ω − (1) | − ω ( x ) | > and M − ω | F X \ ω − is a linear homeomorphism. roof. First, let us show k ω k ∞ ≤
1. Assume that | ω ( x ) | >
1. Let f ∈ F be such that f ( x ) = 0. Then (cid:12)(cid:12)(cid:12)(cid:12)(cid:20) n n P k =1 M kω f (cid:21) ( x ) (cid:12)(cid:12)(cid:12)(cid:12) = | ω n ( x ) f ( x ) | n →∞ −−−→ + ∞ , which contradicts the factthat the sequence (cid:26) n n P k =1 M kω f (cid:27) n ∈ N converges in F .Since ω is continuous due to Proposition 2.2 it follows that ω − (1) is closed. Assumethat ω ( x ) = 1. Then, M ∗ ω x F = x F , and so from the Mean Ergodic Theorem there is f ∈ F such that M ω f = f and f ( x ) = h f, x F i 6 = 0. Hence, from Lemma 2.3 the set ω − (1) is open, and therefore clopen. Additionally, M − ω | F X \ ω − is injective since 1 − ω does not vanish on X \ ω − (1).Next, Mean Ergodic Theorem implies that F is decomposed as Ker ( Id F − M ω ) ⊕ ( Id F − M ω ) F . Since Id F − M ω = M − ω it follows that Ker ( Id F − M ω ) = F ω − (1) , while( Id F − M ω ) F ⊂ F X \ ω − (1) . Finally, as F ω − (1) ∩ F X \ ω − (1) = { } , we conclude that F ∼ = F ω − (1) ⊕ F X \ ω − (1) and M − ω F = M − ω F X \ ω − (1) = F X \ ω − (1) .If M ω is uniformly mean ergodic, it is mean ergodic, and so replacing X with a clopenset X \ ω − (1), and F with F X \ ω − (1) we may assume that Id F − M ω = M − ω is injectiveon F with a dense range. Moreover, from the Mean Ergodic Theorem Id F − M ω has aclosed range, and so M − ω is a linear homeomorphism. As M − − ω = M − ω it follows that − ω is a bounded function, from where inf x ∈ X | − ω ( x ) | > Corollary 3.3.
Assume that X is connected and F is -independent. If ω : X → C is a non-constant function such that M ω is a (uniformly) mean ergodic operator on F ,then k ω k ∞ ≤ , ω ( X ) , ( ω ( X ) ) and M − ω is injective and has a dense range (isa linear homeomorphism). Furthermore, n n P k =1 M kω converges to in the strong operator(norm) topology.Remark . Note that every weakly compact power-bounded operator satisfies the con-dition (ii) of the Mean Ergodic theorem, and so it is mean ergodic. Hence, if k M ω k ≤ ω − (1) is clopen, from part (i) of Theorem 3.2. This is analternative proof of Theorem 2.4. Remark . The converse of Theorem 3.2 is the following: if M ω is a power-boundedoperator on F , F ∼ = F ω − (1) ⊕ F X \ ω − (1) , and M − ω | F X \ ω − has a dense range (is asurjection), then M ω is (uniformly) mean ergodic.Note that unlike the mean ergodicity the condition of uniform mean ergodicity of anoperator on F only depends on the properties of the operator as an element of L ( F )with no reference to F itself. Hence, it is plausible to expect a simple characterization ofuniform mean ergodicity in the NSCF’s whose multiplier algebra has an explicit topology.Let us consider a class of such NSCF’s. We will say that F has a quasi-monotone normif there is α ≥ k f k ≤ α k g k for every f, g ∈ F with | f | ≤ | g | . It is easy tosee that in this case k M ω k ≤ α k ω k ∞ , for every multiplier ω of F . If moreover k ω k ∞ ≤ M ω is power-bounded. Another useful property is given in the following proposition.9 roposition 3.6. If F has a quasi-monotone norm, then the set of its bounded continuousmultipliers is closed in C ∞ ( X ) .Proof. Let { ω n } n ∈ N be a sequence of bounded continuous functions that are multipliersof F and let ω ∈ C ∞ ( X ) be such that k ω − ω n k ∞ n →∞ −−−→
0. Let f ∈ F . Since themultiplier norm does not exceed α k · k ∞ , for some α ≥
1, it follows that { ω n f } n ∈ N ⊂ F is a Cauchy sequence, and so there is g ∈ F such that ω n f n →∞ −−−→ g in F . Since the latteris a NSCF it follows that g = ωf , and since f was chosen arbitrarily, we conclude that ω ∈ M ult ( F ).Now we can state a sufficient condition for uniform mean ergodicity of a MO on aBSCF with a quasi-monotone norm. Proposition 3.7.
Assume that F has a quasi-monotone norm and let ω be a continuousmultiplier of F . If k ω k ∞ ≤ and inf x ∈ X \ ω − (1) | − ω ( x ) | > , then M ω is uniformly meanergodic.Proof. Let U = ω − (1), which is closed, and let β = inf x ∈ X \ U | − ω ( x ) | >
0. Then U = ω − ( B (1 , β )), and so U is open and therefore clopen. Hence, U is continuous.For n ∈ N we have ω n ( x ) − U ( x ) = 0 if x ∈ U , and ω n ( x ) − U ( x ) = ω ( x )1 − ω ( x ) 1 − ω ( x ) n n , if x ∈ X \ U . In both cases | ω n ( x ) − U ( x ) | ≤ β n n →∞ −−−→
0, and so ω n n →∞ −−−→ U in C ∞ ( X ).Then, from Proposition 3.6 U is a multiplier of F . Since n n P k =1 M kω = M ω n , and the normis quasi-monotone we conclude that n n P k =1 M kω n →∞ −−−→ M U in L ( F ). Thus, M ω is uniformlymean ergodic.In the case when X is compact and ω − (1) is clopen, X \ ω − (1) is compact, and soinf x ∈ X \ ω − (1) | − ω ( x ) | > Corollary 3.8.
Assume that X is compact and F is -independent and has a quasi-monotone norm. Then MO on F is uniformly mean ergodic if and only if it is meanergodic.Example . Assume that X is a bounded domain in C n . Since every MO on A ( X )is also a MO on A (cid:0) X (cid:1) , and the latter is a 1-independent BSCF over a compact space X with a quasi-monotone norm, every mean ergodic MO on A ( X ) is in fact uniformlymean ergodic. The same argument works for any uniform algebra (see [14]). Remark . If X is connected, and F has a quasi-monotone norm and is such that everymean ergodic MO is uniformly mean ergodic, then there is no weakly compact MO’s on F other than possibly λId F , λ ∈ C . If ω is non-constant and such that M ω is weaklycompact, then M ωλ k ω k∞ is power-bounded due to quasi-monotonicity of the norm, for every λ ∈ T . Moreover, M ωλ k ω k∞ is mean ergodic, according to Remark 3.4, and so it is uniformlymean ergodic, from our assumption. From Corollary 3.3 therefore λ k ω k ∞ ω ( X ), andsince λ was chosen arbitrary, we get that k ω k ∞ < k ω k ∞ . Contradiction.10et us now discuss some sufficient conditions for mean ergodicity. Assume that k ω k ∞ ≤
1. We know that n n P k =1 M kω f converges pointwise to ω − (1) f , for every f ∈ F .Hence, if additionally M ω is power-bounded, ω − (1) is clopen, and every pointwise con-vergent bounded sequence in F converges weakly, M ω satisfies the condition (ii) of theMean Ergodic theorem, and so is mean ergodic. Let us explore how to guarantee theseconditions in a tangible way. First, recall that if F has a quasi-monotone norm, then M ω is power-bounded. It is also easy to see that if F is an algebra with submultiplicativenorm, the power-boundedness follows from k ω k ≤
1. Applying Grothendieck comple-tion theorem (see [7, III.6, Theorem 1]) to F with pointwise topology and the collectionof closed bounded sets in F , it follows that pointwise and weak topologies coincide onbounded subsets of F if and only if F ∗ = span { x F , x ∈ X } . In turn, this happens when-ever F ∗∗ is a BSCF such that J F ∗∗ = J ∗∗ F . Indeed, injectivity of J ∗∗ F can be expressed by { F ∗∗ } = Ker J ∗∗ F = { x F , x ∈ X } ⊥ , which (in the duality of F ∗ and F ∗∗ ) is equivalent to F ∗ = { F ∗∗ } ⊥ = { x F , x ∈ X } ⊥⊥ = span { x F , x ∈ X } . Example . If v is continuous on X , the spaces C v considered in [4] are BSCF’s withquasi-monotone norms, and moreover every pointwise convergent bounded sequence isweakly convergent. Hence, M ω is mean ergodic on C v as soon as k ω k ∞ ≤ ω − (1)is clopen. Similarly, if X is a domain in C n , the spaces H v considered in [5] are BSCF’swith quasi-monotone norms, and moreover under some mild conditions ( H v ) ∗∗ = H ∞ v with J H ∞ v = J ∗∗H v (see [8] and the reference therein). Hence, in this case M ω is meanergodic on H v as soon as ω ∈ H ( X ) satisfies k ω k ∞ ≤ F is an algebra, then every element of F is a multiplier of F . It is possibleto show that the set of all multipliers of a BSCF is itself a BSCF, and so from the ClosedGraph theorem, F is continuously included into the space of its multipliers. It is easy tosee that the norm ||| · ||| = α k · k on F is submultiplicative, where α be the norm of theinclusion. Using this observation and similar arguments as in the preceding example onecan obtain a sufficient condition of mean ergodicity of MO’s on Little Lipschitz spaces(see [20]).Let us conclude this article by recovering a result from [4]. We will need the followingauxiliary fact. Lemma 3.12.
Let F be a -independent NSCF over X . If { x n } n ∈ N ⊂ X , then F = n f ∈ F , lim n →∞ f ( x n ) k x n F k = 0 o is a closed subspace of F . Moreover, if ω : X → C is a multiplierof F such that lim n →∞ ω ( x n ) = 0 , then M ω F ⊂ F .Proof. For every f ∈ F and ε > g ∈ F such that k f − g k < ε . Thenlim sup n →∞ | f ( x n ) |k x n F k ≤ lim sup n →∞ | g ( x n ) |k x n F k + lim sup n →∞ | f ( x n ) − g ( x n ) |k x n F k ≤ ε. As ε was chosen arbitrarily we conclude that lim n →∞ f ( x n ) k x n F k = 0, and so f ∈ F .11o prove the second statement, take f ∈ F . Then | f ( x n ) |k x n F k ≤ k f k , for every n ∈ N ,from where lim n →∞ [ M ω f ]( x n ) k x n F k = lim n →∞ ω ( x n ) f ( x n ) k x n F k = 0. Proposition 3.13.
Assume that X is Tychonoff and u : X → (0 , + ∞ ) is upper semi-continuous. A multiplication operator is uniformly mean ergodic on C ∞ u if and only if itis mean ergodic.Proof. First, let us show that if in the statement of the lemma { x n } n ∈ N is a discretesubset of X (i.e. discrete in the induced topology), and F = C ∞ u , then the F = C ∞ u .We will construct a sequence { V n } n ∈ N of open sets, such that x n ∈ V n , for every n ∈ N , and V j ∩ V k = ∅ , for distinct j, k ∈ N . Assume that V , ..., V n are chosen (if n = 0nothing is chosen yet). Since { x n } n ∈ N is discrete, there is an open set U n +1 such that U n +1 ∩ { x n } n ∈ N = { x n +1 } . Since x n +1 belongs to U n +1 , but none of V , ..., V n , it followsthat W n +1 = U n +1 \ n S k =1 V k is an open neighborhood of x n +1 . Hence, choose V n +1 to be anopen neighborhood of x n +1 such that V n +1 ⊂ W n +1 .Let w n : X → [0 , + ∞ ] be defined as u on V n and as 0 on X \ V n . Since u isupper semi-continuous and non-vanishing it follows that w n is lower semi-continuouson X . Hence, there is f n ∈ C ∞ ( X ) such that | f n | ≤ w n and f n ( x n ) > w n ( x n ).As f n vanishes outside of V n , and (cid:8) V n (cid:9) n ∈ N are disjoint, the sum f = P n ∈ N f n is well-defined and continuous. Moreover, | f | ≤ u and f ( x n ) = f n ( x n ) > u ( x n ) . Hence,lim sup n →∞ | f ( x n ) |k x n C∞ u k = lim sup n →∞ | f ( x n ) u ( x n ) | ≥ , and so f ∈ C ∞ u \ F .Assume that M ω is mean ergodic. Replacing X with a clopen set X \ ω − (1) if needed,without loss of generality υ = 1 − ω does not vanish. Assume that inf x ∈ X | υ ( x ) | = 0 andconstruct the sequence { x n } n ∈ N ⊂ X as follows. Fix arbitrary x , and if x , ..., x n arealready constructed, choose x n +1 to be such that | υ ( x n +1 ) | < | υ ( x n ) | . It is easy tosee that { x n } n ∈ N is a discrete subset. Hence, M υ C ∞ u ⊂ n f ∈ C ∞ u , lim n →∞ f ( x n ) u ( x n ) = 0 o ,which is a closed proper subspace of C ∞ u .Thus, M − ω does not have a dense range, which contradicts mean ergodicity of M ω ,according to Theorem 3.2. Therefore, inf x ∈ X | − ω ( x ) | >
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