Strongly omnipresent operators: general conditions and applications to composition operators

STRONGLY OMNIPRESENT OPERATORS: GENERAL CONDITIONS AND APPLICATIONS TO COMPOSITION OPERATORS L. BERNAL–GONZ´ ALEZ, M. C. CALDER´ ON–MORENO∗ and K.-G. GROSSE-ERDMANN Abstract This paper studies the concept of strongly omnipresent operators that was recently introduced by the first two authors. An operator Ton the space H(G) of holomorphic functions on a complex domain Gis called strongly omnipresent whenever the set of T–monsters is residual in H(G), and a T–monster is a function fsuch that T f exhibits an extremely ‘wild’ behaviour near the boundary. We obtain sufficient conditions under which an operator is strongly omnipresent, in particular, we show that every onto linear operator is strongly omnipresent. Using these criteria we completely characterize strongly omnipresent composition and multiplication operators. 2000 Mathematics subject classification: primary 30E10; secondary 30H05, 46E10, 47B38. Key words and phrases: Holomorphic function, T–monster, residual set, strongly omnipresent operator, dense range, locally dense range, locally stable operator, composition operator, leftcomposition operator, multiplication operator. 1 Introduction Inspired by the notion of holomorphic monsters as introduced and developed by Luh [Lu1] and the third author [Gr, Kapitel 3] (see also [Lu2, LMM, Sc]), the first two authors have recently introduced the concept of T–monsters [BC], which is associated to a (not necessarily linear) operator Ton the space H(G) of holomorphic functions on a domain Gin C . Roughly speaking, a T–monster is a holomorphic function whose image under Thas an extremely ‘wild’ behaviour near the boundary. In [BC] the concept was defined for G6= C in order that the finite boundary be non-empty, but the interested reader can easily check, using chordal distances, that all proofs can be adapted to the case where the boundary point under consideration is the point of infinity. Consequently, we redefine the concept of T–monsters and its ∗The work of the first two authors has been partially supported by DGES grant PB96–1348 and the Junta de Andaluc´ıa. 0 associated notion of strongly omnipresent operators [BC, Section 2] (see also [Be] for the weaker concept of omnipresent operators) in the following way. Throughout this paper, Gwill be a domain in C and ∂G will denote its boundary taken in the extended complex plane C ∞= C ∪ {∞}. By D we denote the open unit disk. An operator always refers to a continuous (not necessarily linear) mapping. Definition 1.1 ([BC]).(a) A function f∈H(G) is a holomorphic monster if it satisfies the following universality property: (U) for each g∈H( D ) and each t∈∂G there exists a sequence (τn) of affine linear transformations with τn(z)→t(n→ ∞) uniformly on D and τn( D )⊂G(n∈ N ) such that f(τn(z)) →g(z) as n→ ∞ locally uniformly in D . (b) Let T:H(G)→H(G) be an operator. Then a function f∈H(G) is a T–monster if Tf is a holomorphic monster. The set of T–monsters is denoted by M(T). It is not difficult to see that in the case when the point of infinity is an isolated point in ∂G this notion, in general, is strictly stronger than the one given in [BC]; this is so, for example, if Tis the identity operator. See below for a comparison with Luh’s holomorphic monsters [Lu1]. For the closely related notion of strongly omnipresent operators we need some more notation. We denote by O(∂G) = {V⊂ C ∞:Vis open and V∩∂G 6=∅} the set of all open subsets of C ∞that meet the boundary of G. If A⊂ C then Arepresents the closure of A,||f||A:= supz∈A|f(z)|, where fis a complex function defined on A, and LT(A) is the set of all affine linear transformations τ,τ(z) = az +b, such that τ( D )⊂A. Definition 1.2 ([BC]).An operator T:H(G)→H(G) is strongly omnipresent if for all g∈H( D ), ε > 0, r∈(0,1) and V∈O(∂G) the set U(T, g, ε, r, V ) := {f∈H(G) : there exists some τ∈LT(V∩G) such that ||(Tf)◦τ−g||r D < ε} is dense in H(G). Again, in contrast to [BC] we have here allowed the point of infinity as a boundary point. As in [BC, Theorem 2.2] it is easy to prove that Tis strongly omnipresent if 1 and only if the set M(T)of T–monsters is residual, i.e., its complement in H(G) is of first category. As a consequence, every new strongly omnipresent operator Tyields a wealth of new universal functions: there is then a residual set of holomorphic functions fso that each function Tf is a holomorphic monster, that is, each Tf satisfies the universality property (U) stated in Definition 1.1. Examples of strongly omnipresent operators are provided in [BC, Sections 3, 4]. Specifically, if Φ(z) = P∞ j=0 ajzjis a non-zero entire function of subexponential type then the associated linear differential operator Φ(D) = P∞ j=0 ajDjon H(G) is strongly omnipresent. For G= C this result even holds for all entire functions Φ of exponential type. Here Dis the differentiation operator Df =f0,D0=Iis the identity operator and Dj+1 =D◦Dj. Furthermore, if Gis a simply connected domain, a∈G,λ∈ C and his a non-zero entire function of exponential type then the integral operator T on H(G) defined by Tf(z) = λf(z) + Zz ah(z−t)f(t)dt (z∈G) is strongly omnipresent. In particular, if Ψ(z) = P∞ j=0 ajzjis any non-zero function that is holomorphic at 0 then the corresponding linear antidifferential operator Ψ(D−1 a) = P∞ j=0 ajD−j aon H(G) is strongly omnipresent. Here D0 a=Iand, for each j∈ N ,D−j af(f∈H(G)) denotes the unique antiderivative Fof fof order jsuch that F(k)(a) = 0 (k∈ {0,1, ..., j −1}). We note that the holomorphic monsters in the sense of Luh [Lu1] are holomorphic functions that are simultaneously Dj– and D−j a–monsters for all j∈ N 0. Since the intersection of countably many residual sets is again residual, the existence of Luhmonsters is thus a direct consequence of the strong omnipresence of each of the operators Djand D−j a,j∈ N 0. Our aim in this paper is twofold. In Section 2 we derive conditions that guarantee that an operator is strongly omnipresent. This will be done in various ways. First we show how to construct new strongly omnipresent operators from known ones. As an application we will see that every onto linear operator is strongly omnipresent. Next we study the problem under which conditions the existence of a single T-monster suffices to make Tstrongly omnipresent. Finally we derive some workable conditions under which a general operator is strongly omnipresent. In Section 3 we apply these results to furnish new examples of strongly omnipresent operators that are substantially different from differential and antidifferential operators; specifically, we characterize the strongly omnipresent (left- and right-) composition operators and the strongly omnipresent multiplication operators. 2 2 Looking for monsters We begin our investigation into the existence of monsters by constructing new strongly omnipresent operators from known ones. Before we do this we note the following two facts. Let Tbe an operator on H(G), ε > 0, g∈H( D ), r∈(0,1) and V∈O(∂G). Then we have: 1. U(T, g, ε, r, V ) = {f∈H(G) : there exists some non-constant τ∈LT(V∩ G) such that ||(Tf)◦τ−g||r D < ε}={f∈H(G) : there exists some nonconstant τ∈LT(V∩G) such that ||Tf −g◦τ−1||τ(r D )< ε}. The first equality is accomplished by a simple continuity argument. 2. M(T) = \ g,ε,r,V U(T, g, ε, r, V ), where, in fact, it suffices to only consider countable dense subsets of g, ε, r and a suitable collection of countably many sets V that cover ∂G; in addition, each set U(T, g, ε, r, V ) is open. Hence M(T) is always a Gδ–subset. Consequently it is residual if and only if it is dense, and if and only if Tis strongly omnipresent. For details see the proof of Theorem 2.2 in [BC]. To start with, we state without proof the following result, which follows trivially from the definition of T–monster. Proposition 2.1. Let T, S :H(G)→H(G)be operators. Then we have S−1(M(T)) = M(TS). In particular, (a) if M(T)6=∅and Sis onto then M(TS)6=∅; (b) if M(TS)6=∅then M(T)6=∅. Theorem 2.2. Let T, S :H(G)→H(G)be operators such that Tis strongly omnipresent and Sis linear and onto. Then TS is strongly omnipresent. Proof. We have to prove that M(TS) is residual, hence, by the preceding proposition, that S−1(M(T)) is residual. But M(T) is always a Gδ–subset, so S−1(M(T)) is also aGδ–subset because Sis continuous. It remains to see that S−1(M(T)) is dense. Since M(T) is dense, given a non-empty open subset Ain H(G), we obtain that M(T)∩SA 6=∅because SA is open due to the Open Mapping Theorem (recall that Sis linear and onto and that H(G) is an F-space). Hence S−1(M(T))∩A6=∅. Thus, S−1(M(T)) is dense, as required. Corollary 2.3. Every onto linear operator on H(G)is strongly omnipresent. 3 Proof. Apply Theorem 2.2 to T=I, the identity operator, which is strongly omnipresent: take Φ(z)≡zin the example given in the introduction. For example, we know that for each N∈ N the antidifferentiation operator D−N a of order Nis strongly omnipresent on H(G) (take Ψ(z)≡zNin the example given in the Introduction), where Gis a simply connected domain and ais a fixed point in G. Since the differentiation operator D(and so DN) is onto on H(G) due to simple connectedness we can conclude that the operator RN,a on H(G) given by RN,af(z) := f(z)− N−1 X j=0 f(j)(a) j!(z−a)j, that is, the value at zof Taylor’s remainder of order Nof fat a, is strongly omnipresent. Indeed, take T=D−N aand S=DNin Theorem 2.2. Note that neither D−N anor RN,a is onto; they do not even have dense range. In particular we obtain the following application in the case G= C . Corollary 2.4. There is an entire function f(z) = ∞ X n=0 ajzjsuch that each of its Taylor series remainders RNf(z) := ∞ X j=N ajzj(N∈ N 0) is a holomorphic monster. It is evident that the sum T+Sof two strongly omnipresent operators need not be strongly omnipresent: take, for instance, T=I, S =−I. On the other hand, a non-zero multiple λT,λ6= 0, of a strongly omnipresent operator Tis trivially strongly omnipresent. The next result shows that if Tis strongly omnipresent and S is an operator that behaves well near the boundary then their sum T+Sgenerates a strongly omnipresent operator, and a similar result is true for the product T·Sgiven by (T·S)f=Tf ·Sf. Theorem 2.5. Let T, S be two operators on H(G). Assume that there exists a dense subset Γin ∂G such that, for every t∈Γand for every f∈H(G), there exists lim z→t(Sf)(z)∈ C (respectively C \ {0}). Then M(T)⊂ M(T+S) (respectively M(T·S)). In particular, T+S(respectively T·S)is strongly omnipresent if Tis. 4 Proof. Using Lemma 2.1 of [BC], the fact that M(T)⊂ M(T+S) (respectively M(T)⊂ M(T·S)) is straightforward from the definition of monster. The details are left to the reader. As an example of an application of Theorem 2.5 we may consider again the Taylor remainder operator RN,a (given after Corollary 2.3) acting on H(G), where G is any domain such that the point of infinity is not an isolated point of ∂G. Indeed, we can write RN,a =T+Swith T=I, the identity operator, and Sf(z) = −PN−1 j=0 f(j)(a) j!(z−a)j. It is evident that Sis ‘well-behaved’ on Γ := (∂G)\ {∞}. The next result provides us with a condition that guarantees the existence of a residual set of monsters by assuming the existence of at least one monster. For future reference, we isolate this condition and introduce the notion of (local) stability of an operator. Definition 2.1. Let T:H(G)→H(G) be an operator. (a) Let Kbe a compact subset of Gand Ba closed ball contained in G. Then Tis K-stable in Bif for each f∈H(G) and ε > 0 there exist a closed ball B0⊂G\K and δ > 0 such that for all g∈H(G) kf−gkB0< δ implies that kTf −TgkB< ε. (b) We say that Tis locally stable near ∂G if for each compact subset Kof Gthere exists a compact subset Mof Gsuch that for each closed ball B⊂G\M,Tis K-stable in B. (c) We say that Tis somewhere locally stable near ∂G if for each compact subset K of Gand each V∈O(∂G) there exists a closed ball B⊂V∩Gsuch that Tis K-stable in B. For the sake of brevity we will usually take the qualification ‘near ∂G’ for granted. It is clear that every locally stable operator is somewhere locally stable. In fact, every locally stable operator Thas the property that for each compact subset Kof Gand each t∈∂G there is an open neighbourhood Vof tsuch that for each closed ball B⊂V∩G,Tis K-stable in B. By using Cauchy’s integral formula for derivatives, it is straightforward to verify that each differential operator Φ(D), where Φ is an entire function of subexponential type, is locally stable; in fact we can always take concentric closed balls B,B0with radius(B)<radius(B0). Another example of a locally stable operator is the rotation operator Rα(α∈[0,2π)) on H( D ) defined by Rαf(z) = f(zeiα). Further examples will be given in Section 3. 5 Theorem 2.6. Let Tbe an operator on H(G)that is locally stable near ∂G. If M(T)6=∅then Tis strongly omnipresent. Proof. Fix g∈H( D ), ε > 0, r∈(0,1) and V∈O(∂G). We are going to show that U(T, g, ε, r, V ) is dense in H(G). To see this, fix a basic open subset D(h, K, ε1) = {f∈H(G) : kf−hkK< ε1} of H(G), where Kis a compact subset of Gsuch that each connected component of C ∞\Kcontains at least one connected component of C ∞\G,h∈H(G) and ε1>0. Now let fbe a fixed T–monster. Then there exists a non-constant affine linear transformation τ∈LT(G∩V) such that ||Tf −g◦τ−1||B< ε/2,(1) where B:= τ(r D ). We can assume that B⊂G\M, where Mis the compact subset of Ggiven by the definition of local stability. Hence there exist a closed ball B0⊂G\K and δ > 0 such that, for all ϕ∈H(G), kϕ−fkB0< δ implies that kTϕ −TfkB< ε/2.(2) Consider the compact set L:= K∪B0. Then each connected component of the complement of Lcontains at least one component of the complement of Gbecause K has this property and K∩B0=∅. Pick open subsets G1, G2⊂Gwith G1∩G2=∅ and K⊂G1, B0⊂G2. Denote G0=G1∪G2. Hence G0is open and L⊂G0⊂G. Define the function F:G0→ C by F(z) = (h(z) if z∈G1 f(z) if z∈G2. Then F∈H(G0) and an application of Runge’s theorem [Ru, Chapter 13] yields the existence of a rational function f1with poles outside Gsuch that ||f1−F||L<min{δ, ε1}. Thus, f1∈H(G) with ||f1−h||K< ε1(3) and ||f1−f||B0< δ. From (2) we obtain ||Tf1−Tf||B< ε/2.(4) 6 Then (1) and (4) lead us to ||Tf1−g◦τ−1||B< ε. (5) Summarizing, (3) and (5) tell us that f1∈U(T, g, ε, r, V )∩D(h, K, ε1). Consequently, U(T, g, ε, r, V ) is dense, as required. We state another condition under which the existence of a single monster guarantees the existence of a large supply of monsters: We assume that, on a dense set of functions, Tis well-behaved near the boundary. This time linearity of Tis needed. Theorem 2.7. Let Tbe a linear operator on H(G)with M(T)6=∅. If there exists a dense subset Din H(G)with the property that for each h∈ D there is a dense subset Γhin ∂G such that, for all t∈Γh, there exists lim z→t(Th)(z)∈ C then Tis strongly omnipresent. Proof. Pick a T–monster f. Then f+Dis dense in H(G). Fix h∈ D and consider its corresponding set Γh⊂∂G. Given t∈Γhand g∈H( D ), define the function g1(z) := g(z)−L(t), where L(t) := limz→t(Th)(z)∈ C . Then g1∈H( D ) and there exist two sequences of complex numbers (an),(bn) with anz+bn→t(n→ ∞) uniformly on D and anz+bn∈Gfor all n∈ N and all z∈ D such that (Tf)(anz+bn)→g1(z) (n→ ∞) locally uniformly in D . But we also have that (Th)(anz+bn)→L(t) (n→ ∞) locally uniformly in D (in fact, uniformly on the whole D ). Therefore, by linearity, (T(f+h))(anz+bn)→g(z) (n→ ∞) in the same manner. In view of Lemma 2.1 of [BC] this implies that f+D ⊂ M(T), so that M(T) is dense. For instance, the condition in the above theorem is satisfied by a differential operator Φ(D) and by a finite order antidifferential operator Ψ(D−1) whenever Gis a simply connected domain with G6= C : just let Dbe the set of all polynomials and let Γhbe the finite boundary of Gfor all h∈ D. Although the last two theorems give mild conditions under which M(T)6=∅implies that Tis strongly omnipresent the following example shows that this implication does not hold in general. Recall that Idenotes the identity operator. 7 Example 2.8. Consider the operator T:H( D )→H( D ) given by Tf(z) = ψ z 1 + |f(0)|!(z∈ D ), where ψis a fixed holomorphic monster in D . Then fis a T-monster if and only iff(0) = 0. Hence T-monsters exist, but they only form a set of first category. This example gives a partial solution to a problem posed in [BC]. Unfortunately the problem remains open for linear operators: If M(T)6=∅, is Talways strongly omnipresent? Next, we want to derive practicable conditions on an operator that guarantee its strong omnipresence. Here the range, or rather the local ranges, of Twill play an important role. This leads us to the following definition. Definition 2.2. Let T:H(G)→H(G) be an operator. (a) Let U⊂Gbe an open ball. Then Thas dense range in Uif the operator TU:H(G)→H(U) f7→ TUf= (Tf)|U has dense range. (b) We say that Thas locally dense range near ∂G if there exists a compact subset Mof Gsuch that for each open ball U⊂G\M,Thas dense range in U. (c) We say that Thas somewhere locally dense range near ∂G if for each V∈O(∂G) there exists an open ball U⊂V∩Gsuch that Thas dense range in U. As before, we will usually take the qualification ‘near ∂G’ for granted. It is clear that every operator with dense range has locally dense range and that every operator with locally dense range has somewhere locally dense range. Up to now, all the known examples of strongly omnipresent operators satisfy one of the previous properties. For instance, recall that every non-zero differential operator Φ(D) has dense range whenever Gis simply connected: indeed, Φ(D) is onto on the space of entire functions H( C ) (see [Eh, Ma]) and H( C ) is dense in H(G). In fact, in any domain G, Φ(D) always has locally dense range. Also the antidifferential operator D−N ahas locally dense range near the boundary. 8 By Theorem 2.5, taking T=I, the identity operator, and Sthe constant operator with value ψ, we know that Mψis strongly omnipresent if ψextends to a continuous function on the boundary without zeros there. In fact, we can drop this extra condition. Theorem 3.7. If ψis non-zero then the multiplication operator Mψis strongly omnipresent. Proof. For any ψ,Mψis obviously locally stable. Now let V∈O(∂G). If ψ6= 0 there is an open ball U⊂V∩Gsuch that ψhas no zeros in U. Then it is clear that Mψhas dense range in Uby Runge’s theorem. Hence Mψalso has somewhere locally dense range, which implies the result by Corollary 2.12. References [Ah] L. V. Ahlfors, Complex analysis, 3rd. Edition (McGraw–Hill, London, 1979). [Be] L. Bernal–Gonz´alez, ‘Omnipresent holomorphic operators and maximal cluster sets’, Colloq. Math. 63 (1992), 315–322. [BC] L. Bernal–Gonz´alez and M. C. Calder´on–Moreno, ‘Holomorphic T–monsters and strongly omnipresent operators’, J. Approx. Theory 104 (2000), 204–219. [Bi] G. D. Birkhoff, ‘D´emonstration d’un th´eor`eme ´el´ementaire sur les fonctions enti`eres’, C. R. Acad. Sci. Paris 189 (1929), 473–475. [CL] E. F. Collingwood and A. J. Lohwater, The theory of cluster sets (Cambridge University Press, Cambridge, 1966). [Eh] L. Ehrenpreis, ‘Mean periodic functions I’, Amer. J. Math. 77 (1955), 293– 328. [Gr] K.–G. Grosse–Erdmann, ‘Holomorphe Monster und universelle Funktionen’, Mitt. Math. Sem. Giessen 176 (1987). [Lu1] W. Luh, ‘Holomorphic monsters’, J. Approx. Theory 53 (1988), 128–144. [Lu2] W. Luh, ‘Multiply universal holomorphic functions’, J. Approx. Theory 89 (1997), 135–155. [LMM] W. Luh, V. A. Martirosian and J. M¨uller, ‘T–universal functions with lacunary power series’, Acta Sci. Math. (Szeged) 64 (1998), 67-79. 15 [Ma] B. Malgrange, ‘Existence et approximation des solutions des ´equations aux d´eriv´ees partielles et des ´equations de convolution’, Ann. Inst. Fourier (Grenoble) 6(1955/1956), 271–355. [No] K. Noshiro, Cluster sets (Springer–Verlag, Berlin, 1960). [Ru] W. Rudin, Real and Complex Analysis, 2nd. Edition (Tata McGraw–Hill, Faridabad, 1974). [Sc] I. Schneider, ‘Schlichte Funktionen mit universellen Approximationseigenschaften’, Mitt. Math. Sem. Giessen 230 (1997). DEPARTAMENTO DE AN ´ ALISIS MATEM ´ ATICO FACULTAD DE MATEM ´ ATICAS, APDO. 1160 AVENIDA REINA MERCEDES 41080 SEVILLA SPAIN E–mail: lb[email protected] DEPARTAMENTO DE AN ´ ALISIS MATEM ´ ATICO FACULTAD DE MATEM ´ ATICAS, APDO. 1160 AVENIDA REINA MERCEDES 41080 SEVILLA SPAIN E–mail: [email protected] FACHBEREICH MATHEMATIK FERNUNIVERSIT¨ AT HAGEN 58084 HAGEN GERMANY E–mail: [email protected] 16