Strongly omnipresent integral operators

STRONGLY OMNIPRESENT INTEGRAL OPERATORS L. Bernal–Gonz´ alez, M.C. Calder´ on–Moreno and K.–G. Grosse–Erdmann∗ An operator Ton the space H(G) of holomorphic functions on a domain Gis strongly omnipresent whenever there is a residual set of functions f∈ H(G) such that T f exhibits an extremely “wild” behaviour near the boundary. The concept of strong omnipresence was recently introduced by the first two authors. In this paper it is proved that a large class of integral operators including Volterra operators with or without a perturbation by differential operators has this property, completing earlier work about differential and antidifferential operators. 1 Introduction In a recent paper the first two authors have introduced and studied the notion of T-monsters and the related notion of strongly omnipresent operators, see [BC1]. These concepts were motivated by the idea of a holomorphic monster as defined by W. Luh [Lu1] and by a result of the third author [Gr1] that, in the sense of Baire categories, almost every holomorphic function is a monster; see also [Lu2], [Sch], [LMM] and [Gr2, Section 4b]. Let Tbe a (not necessarily linear) operator on the space H(G) of holomorphic functions on a domain Gin C. Then, roughly speaking, a T-monster is a holomorphic function in Gwhose image under Thas an extremely “wild” behaviour near every boundary point of G. Strongly omnipresent operators may be characterized as those operators Tfor which almost every holomorphic function on Gis a T-monster. In another recent paper ∗The work of the first two authors has been partially supported by DGES grant PB96–1348 and the Junta de Andaluc´ıa. the authors have extended the notions of T-monsters and strong omnipresence slightly by allowing the point of infinity as a boundary point of G, see [BCG]. Before we can give the formal definitions we need to fix some notation and terminology. By Dwe denote the open unit ball in C. For a subset Aof C,Arepresents the closure of A,A0its interior and ∂A its boundary; in addition we set kfkA:= sup z∈A |f(z)|, where fis a complex function defined on A. An affine linear transformation τ:C→Cis a mapping of the form τ(z) = az +bwith constants aand b. A closed ball B={z∈C:|z−a| ≤ r}is always assumed to have positive radius r > 0. For such a ball Band a point b∈Bwe denote by A(B) the space of all functions that are continuous in Band holomorphic in B0, and by Ab(B) the subspace of A(B) consisting of all functions with a zero at b. We endow A(B) and Ab(B) with the maximum norm k·k∞:= k·kB. Let Gbe a domain in C. Then H(G) denotes the space of holomorphic functions on G, endowed with the usual topology of local uniform convergence. The boundary ∂G will be taken in the extended complex plane C∞=C∪ {∞}, where C∞is topologized by the chordal metric. 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. In this paper, an operator always refers to a continuous (not necessarily linear) mapping. With this we can recall the following definitions from [BCG], see also [BC1]. Definition 1.1 Let Gbe a domain in Cand T:H(G)→H(G) an operator. Then a function f∈H(G) is called a T-monster if for each g∈H(D) and each t∈∂G there exists a sequence (τn) of affine linear transformations τn(z) = anz+bnwith τn(D)⊂Gfor all n∈N, and τn(z)→t(n→ ∞) uniformly on D such that (Tf)τn(z)→g(z) as n→ ∞ locally uniformly in D. Thus, near any boundary point of Gthe function Tf can approximate any given holomorphic function g∈H(D) on suitable open balls τ(D). For brevity, if A⊂Cwe will write LT(A) for the set of all affine linear transformations τwith τ(D)⊂A. Definition 1.2 Let Gbe a domain in C. Then an operator T:H(G)→H(G) is called 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 k(Tf)◦τ−gkrD< ε} is dense in H(G). It turns out that an operator Tis strongly omnipresent if and only if the set of T-monsters is residual, that is, its complement in H(G) is of first Baire category; in other words: if and only if almost every holomorphic function on Gis a T-monster. This can be accomplished by expressing the set of T-monsters as a suitable countable intersection of sets U(T, g, ε, r, V ), cf. [BC1, Theorem 2.2]. In [BCG, Section 2] the authors have derived conditions under which a general operator is strongly omnipresent. Examples of strongly omnipresent operators have been obtained in [BC1, Sections 3, 4] and [BCG, Sections 2, 3]. Specifically, if Φ(z) = ∞ X j=0 ajzj is a non-zero entire function of subexponential type then the associated linear differential operator Φ(D) = ∞ X j=0 ajDjon H(G) is strongly omnipresent. Here Dis the differentiation operator Df =f0,D0=Iis the identity operator and Dj+1 =D◦Dj; and Φ is said to be of subexponential type if for every ε > 0 there is a constant M=M(ε)>0 such that |Φ(z)| ≤ Meε|z|for all z∈C. If G6=Cis a simply connected domain, a∈Gand Ψ(z) = ∞ X j=0 ajzjis a non-zero function that is holomorphic at 0 then the corresponding linear antidifferential operator Ψ(D−1 a) = ∞ X j=0 ajD−j aon H(G) is strongly omnipresent. Here D0 a=Iand, for j∈N,D−j afdenotes the unique antiderivative Fof fof order jsuch that F(k)(a) = 0 for k= 0,1, ..., j −1. In fact, we have for j∈N D−j af(z) = Zz a f(t)(z−t)j−1 (j−1)! dt (z∈G). Furthermore, the strongly omnipresent (left- or right-) composition operators and multiplication operators have been completely characterized. In the latter case the exact condition is that the multiplication function ψbe non-identically zero on G. From now on, Gwill be a simply connected domain, a∈Ga fixed point and ϕ:G×G→C a function that is holomorphic in both variables. Our aim in this paper is to study the strong omnipresence of operators T:H(G)→H(G) of the form Tf(z) = Sf(z) + Zz a f(t)ϕ(z, t)dt (z∈G), where S:H(G)→H(G) is another (in general non-integral) operator. In Section 2 we obtain a general sufficient condition for Tto be strongly omnipresent. In Section 3 we apply this result to show, among other things, that the sum of a Volterra operator and a finite order differential operator with holomorphic coefficients is always strongly omnipresent. It follows, in particular, that every Volterra operator of the first or second kind is strongly omnipresent, thus generalizing earlier work in [BC1]. Let us emphasize that while in [BCG] Runge’s theorem was sufficient for approximations, Mergelyan’s theorem will be crucial for the results in the present paper. 2 A sufficient condition In [BC1, Theorem 4.2] the first two authors proved that the Volterra operator of the first or second kind on H(G) given by Tf(z) = λf(z) + Zz a f(t)h(z−t)dt (1) is strongly omnipresent, where λ∈C,his a non-zero entire function of exponential type and the integral is taken along any rectifiable curve in Gjoining awith z. The proof of this result has two well-distinguished steps, and the restriction on the integral kernel of Tto be a convolution kernel is only necessary in one of them. If we take this into account then a careful study of the proof enables us to obtain a sufficient condition for the strong omnipresence of integral operators with general kernels. Moreover, we can replace the operator of multiplication with λby more general operators S. Theorem 2.1 Let S:H(G)→H(G)be an operator. Then the operator T:H(G)→H(G) defined by Tf(z) = Sf(z) + Zz a f(t)ϕ(z, t)dt (z∈G) is strongly omnipresent if for each V∈O(∂G)there are closed balls B, B0in V∩Gwith B0⊂Band a point b∈∂B such that (a) the operator Sextends continuously to an operator e S:A(B)→A(B0), (b) the operator e T:Ab(B)→A(B0)defined by e Tf(z) = e Sf(z) + Zz b f(t)ϕ(z, t)dt (z∈B0) has dense range. Proof. The proof is an adaptation of the first half of the proof of Theorem 4.2 in [BC1]. We fix g∈H(D), ε > 0, r∈(0,1) and V∈O(∂G), where we can assume that gis a polynomial. We then have 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 G,h∈H(G) and ε1>0. Then we can find – a compact subset Lof Gwith connected complement that contains aand K, – closed balls B, B0in V∩Gwith B0⊂Band a point b∈∂B that satisfy the hypothesis in the statement of the theorem; by starting with a smaller set V, if necessary, we can assume that B∩L=∅, – an affine linear transformation τwith τ(D) = B0, and – a rectifiable Jordan arc γjoining aand bin Gsuch that γ∩B={b}and such that L∪γ∪Bhas connected complement. Let h1be a function that is continuous on L∪γ, agrees with hon Land satisfies h1(b) = 0. It follows from (b) that there is function h2∈Ab(B) such that e Sh2(z)+ Zz b h2(t)ϕ(z, t)dt−g(τ−1(z)) −Zγ h1(t)ϕ(z, t)dt< ε (z∈B0).(2) By (a) the operator e S:A(B)→A(B0) is continuous, and the same is obviously true for the operators A(B)→A(B0), ψ 7→ Z· b ψ(t)ϕ(·, t)dt and A(γ)→A(B0), ψ 7→ Zγ ψ(t)ϕ(·, t)dt. Hence, by (2), there exists a δ > 0 such that if f∈H(G) satisfies |f(z)−h1(z)|< δ for z∈γ(3) and |f(z)−h2(z)|< δ for z∈B(4) then e Sf(z)+ Zz b f(t)ϕ(z, t)dt−g(τ−1(z)) −Zγ f(t)ϕ(z, t)dt< ε (z∈B0).(5) Now, since the function that agrees with h1on L∪γand with h2on Bis continuous on L0:= L∪γ∪B(note that h1(b) = h2(b) = 0) and holomorphic in the interior of L0, and since L0has connected complement it follows from Mergelyan’s Theorem (see [Rud, Chapter 20]) that there exists a function f∈H(G) that satisfies (3), (4) and |f(z)−h1(z)|< ε1for z∈L. (6) Then falso satisfies (5), which can be rewritten as Tf(z)−g(τ−1(z))< ε (z∈B0). This shows that f∈U(T, g, ε, r, V ); note that τ(D) = B0. In addition it follows from (6) that f∈D(h, K, ε1); recall that h1=hon L⊃K. Thus we have proved that U(T, g, ε, r, V )∩D(h, K, ε1)6=∅, as required. ♦ Using standard arguments, condition (a) in the theorem is implied by the following stronger condition: (a’) for every ε > 0there exists a δ > 0such that for all f, g ∈H(G)we have that kf−gkB< δ implies kSf −SgkB0< ε. Moreover, if Sis a linear operator then the conditions (a) and (a’) are in fact equivalent. This shows that condition (a) is related to the notion of local stability introduced by the authors in [BCG]. For the sake of completeness, we provide the corresponding definition given in that paper. An operator Tis said to be 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, each f∈H(G) and each ε > 0 there exist a closed ball B0⊂G\Kand δ > 0 such that if g∈H(G) and ||f−g||B0< δ then ||Tf −Tg||B< ε. In addition, 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, the operator f∈H(G)7→ (Tf)|U∈H(U) has dense range. Thus, while in [BCG, Section 2] we inferred strong omnipresence of an operator Tfrom local stability and local density of Titself we ask here, roughly, for local stability and local density of two different operators that are related to T. Based on Theorem 2.1 we can now show that several concrete operators are strongly omnipresent. 3 Strongly omnipresent integral operators In the first result of this section we show that the restriction on the kernel of the Volterra operator (1) to be a convolution kernel as considered in [BC1] is not needed for strong omnipresence. Also, we can replace multiplication with a constant λ∈Cby multiplication with a general holomorphic function ψ∈H(G). We first study the case where ψis not identically zero. Theorem 3.1 Assume that ψ∈H(G)is non-zero. Then the operator T:H(G)→H(G) defined by Tf(z) = ψ(z)f(z) + Zz a f(t)ϕ(z, t)dt (z∈G) is strongly omnipresent. Proof. We will show that the following assertion holds: (∗) For every closed ball B⊂Gwith ψ(z)6= 0 for all z∈Band every point b∈∂B the operator e T:Ab(B)→Ab(B) given by e Tf(z) = ψ(z)f(z) + Zz b f(t)ϕ(z, t)dt (z∈B) is onto. Since, as is easy to see, Ab(B) is dense in A(B0) for any closed ball B0contained in the interior of Bit then follows from Theorem 2.1 that Tis strongly omnipresent. Now, since ψ(z)6= 0 for all z∈B, we need only prove that the operator T1: Ab(B)→Ab(B) defined by T1f(z) = f(z) + Zz b f(t)ϕ1(z, t)dt (z∈B) is onto, where ϕ1(z, t) = ϕ(z, t) ψ(z), and this in turn is satisfied whenever T1is invertible. After rotation, translation, normalization and change of sign we can suppose that B=D,b= 1 and that T1=I−K, where Kis the operator Kf(z) = Zz 1 f(t)ϕ1(z, t)dt. We have that T1is invertible on A1(D) if and only if 0 /∈σ(T1), the spectrum of T1. Hence, it suffices to demonstrate that σ(T1) = {1},which, in turn, is equivalent to σ(K) = {0} (that is, that Kis quasi-nilpotent). By Gelfand’s formula for the spectral radius we need to show that kKnk1/n →0 (n→ ∞),(7) where kKk:= sup{kKfk∞:kfk∞≤1}, the norm in the space of linear operators on A1(D). Now, (7) follows as in the classical case of Volterra operators on the real line. Let f∈A1(D) with kfk∞≤1 and set M:= sup{|ϕ1(z, t)|: (z, t)∈D×D}. Then it is clear that, for each z∈D, |Kf(z)| ≤ M|z−1|. By iteration we obtain K2f(z) = Zz 1 (Kf)(t)ϕ1(z, t)dt =Z1 0 (Kf)(1 −u+zu)ϕ1(z, 1−u+zu)(z−1) du, so |K2f(z)| ≤ Z1 0 M|1−u+zu −1| · M|z−1|du ≤M2|z−1|2Z1 0 u du =M2|z−1|2 2. An induction procedure leads us to |Knf(z)| ≤ Mn|z−1|n n!≤(2M)n n!(z∈D,kfk∞≤1, n ∈N). Then kKnk1/n ≤2M (n!)1/n →0 (n→ ∞). Consequently, (7) is satisfied and the proof is finished. ♦ We next want to study the operator Tgiven in Theorem 3.1 when ψ= 0. For this we need the following lemma. Lemma 3.2 Let ϕ:G×G→C,(z, t)7→ ϕ(z, t)be a function that is holomorphic with respect to both variables such that for each number n∈N0and each w∈G, ∂nϕ ∂zn(w, w)=0. Then ϕis identically zero on G×G. Proof. It follows from the hypothesis that for every w∈Gthe function z7→ ϕ(z, w) is holomorphic on Gand has vanishing derivatives of all orders n≥0 at w. Hence it must be identically zero on G, which implies the result since w∈Gis arbitrary. ♦ With this we can complete Theorem 3.1 by considering the case ψ= 0. Of course, we now have to assume that ϕ6= 0.