Boundary-chaotic behaviour of continuous functions under the action of operators

Boundary-chaotic behavior of continuous functions under the action of operators by L. BERNAL-GONZ´ ALEZ and M.C. CALDER´ ON–MORENO∗ Abstract In this paper we introduce two classes of operators on spaces of continuous functions with values in F-spaces under the action of which many functions behave chaotically near the boundary. Several examples, including onto linear operators, left and right composition operators, multiplication operators, and operators with pointwise dense range or with some stability property, are given. This new theory extends one recently developed on spaces of holomorphic functions. Key words and phrases: Omnipresent operator, DI-operator, locally compact space, spaces of continuous function, cluster set, boundary-chaotic function, composition operator, dense range operator, stability near the boundary. 2000 Mathematics Subject Classification: Primary 47B38. Secondary 30D40, 46E10, 54D45. 1 Introduction In this paper we are concerned with the chaotic behavior near the boundary exhibited by certain continuous functions under the action of several kinds of operators. Such chaotic behavior has attracted the attention of many mathematicians during the last decades, mainly in the setting of holomorphic or meromorphic functions on complex domains. Most results obtained in this field are directly or indirectly related to cluster sets. We refer the reader to [8] and [18] for surveys of the classical statements about the matter. Let us introduce the following rather general definition of cluster set. Assume that X,Yare topological spaces and that Gis an open subset of Xwith ∗The authors have been partially supported by Plan Andaluz de Investigaci´on de la Junta de Andaluc´ıa. 0 non-empty boundary ∂G. Let f:G→Ybe a mapping. If t∈∂G, then the cluster set of fat tis defined as the set S(f, t) = \{f(V∩G) : V⊂X, V 3t, V open}, where Adenotes the closure of a subset A. Observe that if thas a denumerable basis of neighborhoods then S(f, t) is the set of all y∈Yfor which there exists a sequence of points (zn)⊂Gwith zn→tand f(zn)→yas n→ ∞. We are interested in the existence of boundary-chaotic functions fin the following sense: A function f:G→Yis said to be boundary-chaotic if and only if each cluster set S(f, t)is maximal, i.e., S(f, t) = Yfor all t∈∂G. For instance, in the case X=Y= C , the complex plane, if fis holomorphic in a punctured neighborhood of tand tis an essential singularity of f, then S(f, t) is maximal. The continuous function f(x) = x−1sin(x−1) is a trivial example for X=Y= R , the real line and G= (0,+∞). If Gis a non-empty open subset of C then we denote, as usual, by H(G) the Fr´echet space (hence a Baire space) of holomorphic functions in G, endowed with the compact-open topology. Through the introduction of the “omnipresent operators” on H(G), the first author showed that most functions (in the sense of Baire) in H(G) together with all their derivatives and antiderivatives are boundary-chaotic, see [1]. This can also be extracted (with different methods) from the results of K.-G. Grosse-Erdmann [10, Kapitel 3] (see also [11, Section 4b]), who in turn improves some statements of W. Luh about the existence of “holomorphic monsters”, see [14]. See also [15], [16] and [19] for further development of the topics. In [1] the omnipresence is also shown for a rather general class of integral operators. The theory created by Luh and Grosse-Erdmann was recently extended by the authors in [3] via the introduction of the “T-monsters” and of the “strongly omnipresent operators” (these are a special case of omnipresent operators). Several examples of this kind of operators, including those of the form Φ(D),Φ(D−1) (where Ddenotes differentiation and Φ denotes an adequate non-zero holomorphic function), are furnished. Additional examples 1 can be found in [5] and [6]. The strongly omnipresent operators are related to certain generalized cluster sets introduced by Luh in [14] for whose definition affine linear transformations z7→ az +bare used; hence, unlike the omnipresent operators, a natural extension to general topological spaces does not seem to be possible for the strong omnipresence. On the other hand, the first author proves in 1995 [2] that, given any nonrelatively compact subset A⊂G, most functions fin H(G) have the property that f(n)(A) is dense in C for every n= 0,1,2, . . .. The second author has recently established that a wide class of differential operators Φ(D) and of integral operators shares the same property [7]. This has led the authors to introduce the concept of “dense-image operator” or, in short, “DI-operator”, see [4]. It happens that an operator on H(G) is omnipresent whenever it is a DI-operator. This concept can as well be defined on more general topological spaces, see Section 2. Our aim in this paper is to study the omnipresent operators and the DIoperators on spaces of continuous functions, as well as to provide several (general or more concrete) examples, see Sections 2–4. We want to point out that while Runge-Mergelyan’s theorems were the natural basic tools to attack the corresponding problems on spaces of holomorphic functions, we cannot use them, of course, in the setting of continuous functions. In addition, observe that in the case Y= C ,G= an open subset of C , the space H(G) is closed (hence non-dense) in C(G, Y ) := {continuous functions f:G→Y}. Consequently, even though an operator Ton C(G, Y ) takes H(G) into itself, we cannot assure from the omnipresence (or from the property DI) of T|H(G) that Titself has the same property. This justifies an independent study for continuous functions. 2 Definitions and preliminary results From now on, Xwill denote a Hausdorff, second countable locally compact topological space, Gwill be an open σ-compact subset of Xwith ∂G 6=∅ and Ywill stand for a separable F-space (= metrizable complete topological vector space). Observe that if ∂G 6=∅then ∅ 6=G6=X. Conversely, under the additional hypothesis of connectedness of X, the latter condition guarantees 2 that the boundary of Gis non-empty. Indeed, if ∂G =∅then G\G=G\G0= ∅, where A0denotes the interior of a subset A. Therefore G=G, so Gis open and closed, which contradicts the connectedness of X. We will promptly need Gto be Hausdorff, locally compact and σ-compact. The first two properties are inherited from X(the second one due to the fact that Gis open), but not the third one. This is the reason why we have to impose that Gbe σ-compact. Indeed, if Ais any non-denumerable set with the discrete topology then Ais, trivially, Hausdorff, locally compact and non-compact, and if X=A∪ {w} is its Alexandroff compactification (see, e.g., [17, Vol. 3, pp. 10–19]) then X is σ-compact (since it is compact), Hausdorff and locally compact, Ais open in X,∂A 6=∅(because ∂A ={w}), but Ais not σ-compact due to nondenumerability. Since our Gis σ-compact, there is a sequence (Kn) of compact subsets with Kn⊂K0 n+1 for all nand G=∪∞ n=1Kn(see, for instance, [12, pp. 325– 326]). From this it is easy to see that if Kis a compact subset of Gthen K⊂Knfor some n. Therefore the same construction given in [9, Chapter 7] and [13, p. 136] can be carried over in order to make the linear space C(G, Y ) an F-space (hence a Baire space) with, for instance, the distance ρ(f, g) = ∞ X n=1 1 2n·maxx∈Knd(f(x), g(x)) 1 + maxx∈Knd(f(x), g(x)), where dis any complete translation-invariant distance on Y. The ρ-convergence is precisely the uniform convergence on compact subsets of G. It is well-known that the family of sets D(g, K, ε) = {f∈C(G, Y ) : d(f(x), g(x)) < ε ∀z∈K},(1) where ε > 0, g∈C(G, Y ) and K⊂Gis compact, is a basis for the topology given on C(G, Y ). An operator on C(G, Y ) always refers to a continuous (not necessarily linear) selfmapping T:C(G, Y )→C(G, Y ). Denote O(∂G) := {V⊂X:Vis open and V∩∂G 6=∅}. Given subsets A⊂X,B⊂Yand an operator Ton C(G, Y ), let us denote R(T, A, B) = {f∈C(G, Y ) : exists a∈A∩Gwith Tf(a)∈B}. Remarks 2.1 (a) It is evident that R(T, A, B)⊂R(T, A0, B0) if A⊂A0and B⊂B0. 3 (b) The set R(T, A, B) is open for all A⊂Xwhenever Bis open. Indeed, we have that R(T, A, B) = [{ϕ−1 a(B) : a∈A∩G}, where ϕais the evaluation mapping ϕa:f∈C(G, Y )7→ T f(a)∈Y, which is clearly continuous. For an operator Ton C(G, Y ) we introduce the followings definitions: We say that Tis omnipresent if and only if R(T, V, W )is dense in C(G, Y ) for every V∈O(∂G)and every non-empty open subset W⊂Y. We say that Tis a DI-operator if and only if R(T, A, W )is dense in C(G, Y )for every non-relatively compact subset Ain Gand every non-empty open subset W⊂Y. Note that each DI-operator is omnipresent, because V∩Gis non-relatively compact in Gfor every V∈O(∂G). Indeed, pick t∈V∩∂G and assume, by way of contradiction, that there is a compact set Kwith V∩G⊂K⊂G. Then (X\K)∩Vis an open subset of Xcontaining t, hence G∩(X\K)∩V6=∅, so V∩G6⊂ K, which is absurd. In Section 4 several examples of omnipresent operators which are not DI-operators will be found. Denote by Ch(T) the set of functions f∈C(G, Y ) such that T f is boundary-chaotic, see Section 1. If A⊂G, we denote by M(T, A) the set M(T, A) = {f∈C(G, Y ) : Tf(A) is dense in Y}. It should be noted that Ch(T) = \R(T, V, W ) (2) and M(T, A) = \R(T, A, W),(3) where Vruns over the members of O(∂G) and Wruns over all non-empty open subsets of Y. In particular, Ch(T) and M(T, A) are Gδsubsets of C(G, Y ). Observe that if Ais relatively compact then g(A) is dense in Yfor no continuous function g. The following proposition shows that under adequate assumptions on Xand Y, the fact “Tis omnipresent” (or “Tis a DI-operator”) 4 means, roughly speaking, that “most functions behave wildly near the boundary under the action of T”. Recall that, in a Baire space, a subset is residual if and only if its complement is of first category. Proposition 2.2 Assume that Tis an operator on C(G, Y ). We have: (a) The operator Tis a DI-operator if and only if M(T, A)is dense for every non-relatively compact subset A⊂Gif and only if M(T, A)is residual for every non-relatively compact subset A⊂G. (b) The operator Tis omnipresent if and only if Ch(T)is dense if and only if Ch(T)is residual. Proof. Since Yis separable we can fix a denumerable open basis (Wn) for Y. Since Xis also second-countable, we can also fix a denumerable open basis for Xand extract from it the sequence (Vj) of members meeting ∂G. Then (a) and (b) follow at once from Remarks 2.1, from the fact that C(G, Y ) is a Baire space and from the equalities (derived from (2)–(3)) M(T, A) = \ n∈ N R(T, A, Wn), Ch(T) = \ j,n∈ N R(T, Vj, Wn). ♦ The next auxiliary extension result will prove very useful for discovering chaotic behavior. Lemma 2.3 Assume that x0∈G,y0∈Yand that Kis a compact subset of Gwith x06∈ K. If g∈C(G, Y )then there exists h∈C(G, Y )such that h(x0) = y0and h(x) = g(x)for all x∈K. Proof. Since Gis Hausdorff, Kis closed. Since Gis Hausdorff and locally compact, it is a Tychonoff space [17, Vol. 2, p. 231], whence there exists a continuous function f:G→ K such that f(x0) = 0 and f(x) = 1 for all x∈K. Here K = R or C is the base field of Y. Since Yis a topological vector space, the mapping h:G→Ygiven by h(x) = f(x)g(x) + (1 −f(x))y0 5 is continuous. A simple glance shows that hsatisfies the required properties. ♦ A more direct proof of the last lemma (using distances to construct the auxiliary function f) can be made by taking into account that X(hence G either) is metrizable, since any Hausdorff locally compact second countable space is metrizable [12, p. 342]. Nevertheless, we will not use this property. To finish this section, we propose the following definitions in order to isolate several conditions which will make an appearance during the next section. Assume that Tis an operator on C(G, Y ). We say that Tis pointwise stable near the boundary if and only if for every compact subset K⊂Gthere exists a compact subset M⊂Gwith the property that for every a∈G\M, every f∈C(G, Y )and every neighborhood Wof Tf(a)there exists a point b∈G\Ksuch that if g∈C(G, Y )and g(b) = f(b) then Tg(a)∈W. The property Tg(a)∈Wcan be expressed, of course, in terms of the distance of Y. The second concept is as follows. We say that Tis somewhere pointwise stable near the boundary if and only if for every compact subset K⊂Gand every V∈O(∂G)there exists a point a∈V∩Gwith the property that for every f∈C(G, Y )and every neighborhood Wof Tf(a)there exists a point b∈G\Ksuch that if g∈C(G, Y )and f(b) = g(b)then Tg(a)∈W. As for an example, the reflection T f(x) = f(−x) (where X:= R , x ∈G:= (−1,1)) is pointwise stable near the boundary. We will give other examples in Section 4. It is easy to see that if an operator Tis pointwise stable near the boundary then it is somewhere pointwise stable near the boundary. Finally, a pair of definitions related to denseness are introduced. We say that Thas pointwise dense range near the boundary whenever there is a compact subset K⊂Gsuch that the set {T f(a) : f∈C(G, Y )}is dense in Yfor every a∈G\K. A corresponding weaker property is the following. 6 We say that Thas somewhere pointwise dense range near the boundary if and only if for each V∈O(∂G)there exists a point a∈V∩Gsuch that {Tf(a) : f∈C(G, Y )}is dense in Y. Trivially, dense range implies pointwise dense range. 3 General theory and some examples Our objectives are to produce DI-operators and omnipresent operators from known others as well as to furnish sufficient conditions for an operator to be DI or omnipresent. We also want to provide concrete examples of these kinds of operators. Therefore, it is natural to wonder whether an easy example is available to start with. Surprisingly, the easiest operator does the job. Theorem 3.1 The identity operator on C(G, Y )is a DI-operator. Hence it is omnipresent. Proof. Fix a non-relatively compact subset Ain Gand a non-empty open subset W⊂Y. Consider the operator Tgiven by T f =fand a basic neighborhood D(g, K, ε) like that in (1). Since R(T, A, W ) = {f∈C(G, Y ) : ∃a∈Awith f(a)∈W}, we have to find a function h∈C(G, Y ) and a point a∈Ain such a way that d(h(x), g(x)) < ε ∀z∈Kand h(a)∈W. Since Ais non-relatively compact, there is a∈(G\K)∩A. Fix any point b∈W. By Lemma 2.3, there exists h∈C(G, Y ) such that h(a) = band h(x) = g(x) for all x∈K, which proves the theorem. ♦ Next, we consider compositions of our operators with other suitable operators. But, before this, observe that if Tand Sare operators on C(G, Y ) then R(TS, A, B) = S−1(R(T, A, B)) (A⊂X, B ⊂Y).(4) Theorem 3.2 Assume that T,Sare operators on C(G, Y ), in such a way that Tis DI (omnipresent) and Sis linear and onto. Then T S is DI (omnipresent, resp.). 7 Proof. The Open Mapping Theorem (recall that C(G, Y ) is a Fr´echet space) tells us that Stakes open sets into open sets, hence S−1(R(T, A, B)) is dense whenever R(T, A, B). Now, equality (4) and the definitions of DI and omnipresent operators prove the assertion. ♦ Corollary 3.3 If Sis an onto linear operator on C(G, Y )then Sis a DI (hence omnipresent) operator. Proof. Combine Theorems 3.1–3.2. ♦ Our next result establishes that if a DI (or omnipresent) operator is perturbed by an operator which is “controlled” near the boundary then a new “wild” operator is obtained, at least when Gis relatively compact in X. This happens, for instance, when Xitself is compact. Theorem 3.4 Suppose that Gis compact. Let Tbe a DI (omnipresent) operator on C(G, Y ). Assume that Sis an operator on C(G, Y )satisfying that, for every f∈C(G, Y )and every t∈∂G, there exists limx→tSf(x)∈Y. Then T+Sis a DI (omnipresent, resp.) operator. Proof. Assume that Tis a DI operator. Fix a non-relatively compact subset A⊂G, a non-empty open subset W⊂Yand a basic neighborhood D(g, K, ε) as in (1). In order that T+Sbe a DI-operator, it must be proved that R(T+S, A, W ) is dense. There exist a vector w∈Yand a neighborhood Uof the origin in Ywith W⊃w+U+U. Since Gis compact and Ais not relatively compact in G,A must be a compact set which is not included in G, so there exists t∈A\G, whence t∈A∩∂G. But there exists y:= limx→tSf(x)∈Y. Then we can find an open subset V⊂Xcontaining tsuch that (Sf)(x)∈y+Uwhenever x∈V. Clearly, tis also in the closure of A∩V, hence A∩Vis not relatively compact in Gbecause t6∈ G. By hypothesis, the set R(T, A ∩V, w −y+U) is dense in C(G, Y ). But if f∈R(T, A∩V, w−y+U) then there exists a∈A∩V with Tf(a)∈w−y+U. Therefore (T+S)f(a)∈w−y+U+y+U=w+U+U⊂W, 8 and, from (3) and (5), M(CϕT, A) = M(T, ϕ(A)).(6) Firstly, we have again by Proposition 2.2 that (a) implies (b) in both (A) and (B). Assume that (b) holds in (A) and fix a non-relatively compact subset A⊂G. Then there is f∈M(Cϕ, A) = M(I, ϕ(A)), where the last equality is due to (6) as applied on the identity operator. Therefore f(ϕ(A)) is dense in Y, which forces ϕ(A) to be non-relatively compact in G, which gives (c). If now (b) holds in (B), then we can pick f∈Ch(Cϕ). Assuming V∈ O(∂G), the set Cϕf(V∩G) = f(ϕ(V∩G)) should be dense in Y, hence ϕ(V∩G) cannot be relatively compact in G, and this is (c) of (B). Again by (6), we get M(Cϕ, A) = M(I, ϕ(A)). Then (c) implies (a) in (A): just combine Proposition 2.2(a) and Theorem 3.1. Finally, starting from (c) of (B), in order to prove (a) we should show that R(Cϕ, V, W ) = R(Cϕ, V ∩G, W) is dense in C(G, Y ) for every V∈O(∂G) and every non-empty open subset W⊂Y. But, by (5), R(Cϕ, V ∩G, W) = R(I, ϕ(V∩G), W ), and this set is dense because Iis DI (not only omnipresent!). ♦ Remarks 4.4 A systematic application of (5) allows us to arrive at the following statements: (i) The operator CϕTis DI for each DI-operator Tif and only if ϕis proper. (ii) The operator CϕTis omnipresent for each DI-operator Tif and only if ϕ(V∩G)is non-relatively compact in Gfor all V∈O(∂G). (iii) The operator CϕTis omnipresent for each omnipresent operator Tif ϕ is “open in the boundary”, that is, given V∈O(∂G)there exists V?∈ O(∂G)with V?∩G⊂ϕ(V∩G). Observe that a new example of a linear, omnipresent, non-DI operator may be extracted from Theorem 4.3. Indeed, X= [0,1], G= [0,1) and Y= R . If ϕ(x) = x|sin( 1 1−x)|, then Cϕgives a suitable operator. Nevertheless, it is not possible to construct a similar example by employing left composition operators, as we can see in the next surprising (and final) theorem. 15 Theorem 4.5 The next seven properties are equivalent: (a) The operator Lαis DI. (b) The operator Lαis omnipresent. (c) The set M(Lα, A)is non-empty for every non-relatively compact subset A⊂G. (d) The set Ch(Lα)is non-empty. (e) The operator LαTis DI for every DI-operator T. (f) The operator LαTis omnipresent for every omnipresent operator T. (g) The function αhas dense range, i.e., α(Y) = Y. Proof. The proof can be accomplished by the interested reader if the following facts are applied: the identity is a DI-operator; every DI-operator is omnipresent; the characterization given in Proposition 2.2; αhas dense range if and only if α−1(W) is non-empty for each non-empty open subset W⊂Y; for every operator Ton C(G, Y ), every A⊂Gand every B⊂Y, R(LαT, A, B) = R(T, A, α−1(B)). 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LUIS BERNAL-GONZ ´ ALEZ MAR´ IA DEL CARMEN CALDER ´ ON-MORENO DEPARTAMENTO DE AN ´ ALISIS MATEM ´ ATICO DEPARTAMENTO DE AN ´ ALISIS MATEM ´ ATICO FACULTAD DE MATEM ´ ATICAS, APDO. 1160 FACULTAD DE MATEM´ ATICAS, APDO. 1160 AVENIDA REINA MERCEDES AVENIDA REINA MERCEDES 41080 SEVILLA, SPAIN 41080 SEVILLA, SPAIN E-mail: lb[email protected] E-mail: [email protected] 18