Hypercyclic sequences of differential and antidifferential operators

TITLE: HYPERCYCLIC SEQUENCES OF DIFFERENTIAL AND ANTIDIFFERENTIAL OPERATORS. AUTHOR: LUIS BERNAL-GONZ´ ALEZ. AFFILIATION: DEPARTAMENTO DE AN´ ALISIS MATEM´ ATICO. FACULTAD DE MATEM´ ATICAS. AVENIDA REINA MERCEDES. APARTADO 1160. 41080 SEVILLA, SPAIN. E-MAIL: lb[email protected]. FOOTNOTES TO THE TITLE: *This work is supported in part by DGICYT grant PB96-1348. 1991 Mathematics Subject Classification: Primary 47B99. Secondary 30E10, 32A07. Key words and phrases: hypercyclic operator, hypercyclic sequence, Fr´echet space, invariant linear manifold, analytic function of several complex variables, Runge domain, infinite order differential and antidifferential operators, zero-free function. 1 ABREVIATED TITLE: HYPERCYCLIC SEQUENCES. NAME AND MAILING ADDRESS OF THE AUTHOR TO WHOM PROOFS SHOULD BE SENT: LUIS BERNAL-GONZ´ ALEZ. DEPARTAMENTO DE AN´ ALISIS MATEM´ ATICO. FACULTAD DE MATEM´ ATICAS. AVENIDA REINA MERCEDES. APARTADO 1160. 41080 SEVILLA, SPAIN. E-MAIL: lb[email protected]. 2 HYPERCYCLIC SEQUENCES OF DIFFERENTIAL AND ANTIDIFFERENTIAL OPERATORS By LUIS BERNAL–GONZ´ ALEZ* Abstract. In this paper, we provide some extensions of earlier results about hypercyclicity of some operators on the Fr´echet space of entire functions of several complex variables. Specifically, we generalize in several directions a theorem about hypercyclicity of certain infinite order linear differential operators with constant coefficients and study the corresponding property for certain kinds of “antidifferential” operators which are introduced in the paper. In addition, the existence of hypercyclic functions for certain sequences of differential operators with additional properties, for instance, boundedness or with some nonvanishing derivatives, is established. 1. INTRODUCTION AND NOTATION In this paper we denote by Nthe set of positive integers, by Cthe field of complex numbers and by N0the set N0=N∪ {0}. Let X, Y be two topological spaces, Tn:X→Y(n∈N) a sequence of continuous mappings and x∈X. Then xis said to be hypercyclic (or universal) for {Tn}if its orbit {Tnx:n∈N} under {Tn}is dense in Y. The sequence {Tn}is hypercyclic whenever it has a hypercyclic element. It is clear that, in order that {Tn}can be hypercyclic, Ymust 3 be separable. If T:X→Xis a continuous selfmapping, then an element x∈Xis said to be hypercyclic for Tif and only if it is hypercyclic for the sequence {Tn}, where Tn=T◦T◦... ◦T(ntimes). Tis hypercyclic when there is a hypercyclic element for T. A subset A⊂Xis invariant under Twhen TA ⊂A. It is evident that xis hypercyclic for Tif and only if there is no proper, closed, invariant subset in Xcontaining x. So, hypercyclicity is connected with the problem of the invariant subspace. If Xis a linear topological space, we say that Tis an operator on X whenever Tis a continuous linear transformation taking Xinto itself. We now furnish a sufficient condition for a sequence {Tn}to be hypercyclic. Its proof is an easy application of the Baire Category Theorem and is left to the reader. Several versions of this result have earlier appeared in [13, Section 2], [14, Section 1], [15, Satz 1.2.2], [16] and [19, Theorem 2.1]. Note that, in a Baire space X, a dense Gδsubset is “very large” in X. A subset A⊂Xis residual if and only if it contains a dense Gδsubset. THEOREM 1. Let Xbe a linear topological space that is a Baire space, Ya metrizable separable (linear topological space, D⊂Xdense in X,E⊂Ydense in Yand Tn:X→Y(n∈N)a countable family of continuous linear mappings satisfying the following condition: For every d∈Dand every e∈Ethere is a sequence {xk} ⊂ X and a subsequence {nk}of positive integers such that xk→dand 4 Tnk(xk)→e(k→ ∞). Then {Tn}has a dense Gδsubset of hypercyclic vectors. The existence of hypercyclic operators on any separable Fr´echet space has been recently proved in [1] (see also [7]). B. Beauzamy [2, 3, 4] has constructed examples of linear operators on Hilbert spaces having dense, invariant linear manifolds all of whose nonzero elements are hypercyclic. P. S. Bourdon [11] proved in 1993 that any hypercyclic operator on a complex Banach space has a dense, invariant linear manifold consisting, except for zero, entirely of hypercyclic vectors. In fact (see [1]) this result holds in a more general setting. We state it for future references. THEOREM 2. Let Tbe a hypercyclic operator on a complex, separable, locally convex space X. Then there is a dense T-invariant linear manifold of Xconsisting entirely, except for zero, of vectors that are hypercyclic for T. Let Gbe a nonempty open subset of CN(N∈N). Gis said to be a domain when, in addition, it is connected. A domain G⊂CNis said to be a Runge domain if each analytic function on Gcan be approximated uniformly by polynomials on every compact subset of G(see [18, pp. 52-59] and [20, Chapter 5]). When N= 1, the Runge domains are precisely the simply connected domains. Denote by H(G), as usual, the Fr´echet space of analytic functions on Gendowed with the compactopen topology. G. D. Birkhoff [10] showed in 1929 that every translation operator 5 τa(that is, τaf(z) = f(z+a), where a∈C\ {0}is fixed) is hypercyclic on the space H(C) and G. R. MacLane [23] obtained the same conclusion in 1952 for the operator of differentiation f7→ f′. G. Godefroy and J. H. Shapiro [14, Section 5] demonstrated in 1991 the following strong generalization of the theorems of Birkhoff and MacLane: THEOREM 3. If Lis an operator on the space H(CN)of entire funcions on CNthat commutes with each of the translation operators τa(a∈CN), and is not a scalar multiple of the identity, then Lhas a dense, invariant vector manifold each of whose non-zero elements is hypercyclic for L. See also [5, 8, 9, 15 and 22] for other generalizations of Birkhoff-MacLane’s theorems. Several works have been made in connection with additional properties imposed to hypercyclicity. For instance, Grosse-Erdmann [16] proved in 1990 that there is no hypercyclic entire function ffor the differentiation operator Dsatisfying max|z|=r|f(z)|=O(er/r1/2) (r→ ∞), while there is a D- hypercyclic entire function fsuch that max|z|=r|f(z)|=O(φ(r)·er/r1/2) (r→ ∞), φ: (0,+∞)→(0,+∞) being a prefixed function such that φ(r)→ ∞ (r→ ∞). G. Herzog [17] showed in 1994 that there is a D-hypercyclic function fsuch that f and f′are zero-free. This result has been recently improved by the author [6], which proves that, if q∈N0and a nonconstant entire function Φ of subexponential type 6 are given, then the set A={f∈H(C) : f(q)and f(q+1) are zero-free}contains a residual subset of Φ(D)-universal functions. The result is sharp in terms of the growth and the type of Φ. In this paper we extend Theorem 3 and the result of the latter paragraph about zero-free derivatives to more general domains and sequences of operators and introduce and study a new kind of operators related to antiderivatives. The existence of bounded hypercyclic functions is established for certain domains. We also provide a rather general “eigenvalue test” in order to prove the hypercyclicity of certain kinds of operators and sequences of operators. 2. DIFFERENTIAL AND ANTIDIFFERENTIAL OPERATORS In order to generalize in Section 4 Godefroy-Shapiro’s result stated in Section 1, we adopt the notation of [14, Section 5] and transcribe some preliminaries from it. For 1 ≤j≤Nlet Djdenote complex partial differentiation with respect to the jth coordinate. A multi-index is an N-tuple p= (p1, ..., pN) of nonnegative integers. Denote |p|=p1+... +pN,p! = p1!·... ·pN!, Dp=Dp1 1◦... ◦DpN N(D0=I= the identity operator) and zp=zp1 1·... ·zpN Nif z= (z1, ..., zN). An entire function Φ(z) = ∑|p|≥0apzpon CNis said to be of exponential type whenever there exist positive constants Aand Bsuch that |Φ(z)| ≤ AeB|z|(z∈CN). This happens if 7 and only if there is R∈(0,+∞) for which |ap| ≤ R|p| p!(|p| ≥ 0). It is shown in [14] that, if Φ is of exponential type, then the mapping Φ(D) = ∑|p|≥0apDpis a well-defined operator on H(CN). Note that if Φ is an entire function and L= Φ(D), then Ln= Φn(D) for all n∈N(Ln= L ◦L◦. . . ◦Lbut Φn= Φ ·Φ·. . . ·Φ, ntimes). Trivially, every linear differential operator with constant coefficients commutes with translations. In [14] it is shown that the operators on H(CN) commuting with translations behave as “infinite order” differential operators. THEOREM 4. Let Lbe an operator on H(CN). The following conditions are equivalent: a) Lcommutes with every translation operator τa(a∈CN). b) Lcommutes with each of the differentiation operators Dk(1 ≤k≤N). c) L= Φ(D), where Φis an entire function on CNof exponential type. Some additional notations and results are needed in order to prove our theorems. If a= (a1, ..., aN)∈CNand r > 0, we denote by D(a, r) the closed polydisc D(a, r) = {z∈CN:|zj−aj| ≤ r, 1≤j≤N}. We consider in CNthe distance d(z, a) = max{|z1−a1|, ..., |zN−aN|}. If gis a function defined on a subset B⊂CN, then ||g||Bwill stand for sup{|g(z)|:z∈B}. We say that an entire func- 8 tion Φ(z) = ∑|p|≥0apzpon CNis of subexponential type whenever, given ε > 0, there exists a positive constant K=K(ε) such that |Φ(z)| ≤ Keε|z|(z∈CN). A straightforward computation with power series and the Cauchy inequalities [18, p. 27] shows that Φ is of subexponential type if and only if, given ε > 0, there is a positive constant A=A(ε) such that |ap| ≤ A·ε|p| p!(|p| ≥ 0). Note that, if N= 1, then Φ is of subexponential type if and only if Φ is either of growth order less than one or of growth order one and growth type zero. Each entire function of subexponential type is obviously of exponential type. THEOREM 5. If G⊂CNis a nonempty open subset and Φ(z) = ∑|p|≥0apzp is an entire function of subexponential type, then the series Φ(D) = ∑|p|≥0apDp defines an operator on H(G). Proof. If G=CN, the result is a particular case of the above considerations. So, we may suppose that G=CN. Fix f∈H(G) and a compact subset K⊂G. Let ε=1 2d(K, CN\G). Then there is A∈(0,+∞) such that |ap| ≤ A·(ε/2)|p| p! (|p| ≥ 0). Fix a point a∈K. The Cauchy formula for derivatives [18, p. 27, Formula 2.2.3] tells that |Dpf(a)| ≤ p!||f||D(a,ε) ε|p|≤p!||f||K1 ε|p|, where K1is the compact set {z:d(z, K)≤ε}. Note that K⊂K1⊂G. Therefore 9 such that lim k→∞ Φnk(a) = 0 for all a∈F1and lim k→∞ Φnk(b) = ∞for all b∈F2. (Q) m(Φn)→ ∞ (n→ ∞) and there is a nonempty open subset B⊂ CNsuch that for every finite subset F⊂Bthere exists a subsequence {nk}of positive integers satisfying limk→∞ Φnk(b) = ∞for all b∈F. (Q’) m(Φn)→ ∞ (n→ ∞) and there is a subset B⊂CNwith at least one finite accumulation point such that for every finite subset F⊂Bthere exists a subsequence {nk}of positive integers satisfying limk→∞ Φnk(b) = ∞for all b∈F. Trivially (P) implies (P’) and (Q) implies (Q’). For instance, the sequence Φn(z) = zn(z∈C;n∈N) satisfies all four properties; the sequence Φn(z) = nnznsatisfies (Q) but does not (P) (nnzn→ ∞ as n→ ∞ for every z∈C\ {0}); the sequence Φn(z) = nenz +zn n2satisfies (P) (take A={z:|z|<1,Re z < 0}and B={z: |z|<1,Re z > 0}) but not (Q). THEOREM 8. Suppose that Gis a Runge domain of CNand Φ,Φn(n∈N) are entire functions on CN. Assume that Φis not a constant and denote Ln= Φn(D) (n∈N). a) Suppose that every Φnis of subexponential (exponential, resp.) type and the sequence {Φn}satisfies (P). Then there is a dense Gδsubset of H(G)(H(CN), resp.) 16 all of whose elements are hypercyclic functions for {Ln}. b) For N= 1 the statement of a) still holds if (P) is changed to (P’). c) Suppose that Φis of subexponential type and let L= Φ(D). Then there is a dense Gδsubset Mof H(G)all of whose elements are hypercyclic functions for L. In addition, Mcontains all nonzero functions of a dense, L-invariant, linear submanifold of H(G). d) Suppose that Lis an operator on H(CN)that commutes with each of the translation operators τa(a∈CN), and is not a scalar multiple of the identity. Then Lhas a dense Gδsubset Mof hypercyclic functions. In addition, Mcontains all nonzero functions of a dense, L-invariant, linear submanifold of H(CN). Proof. a) Firstly, by Theorem 5 and the initial considerations of Section 2, every Lnis an operator defined on H(G) (even on H(CN) if Φnis of exponential type). From now on, Gmay be CNor not. Note that Djea=ajeafor each j∈ {1, ..., N}and each a∈CN, so Dpea=apeafor every multi-index p. Then Lnea= Φn(D)ea= Φn(a)ea(a∈CN, n ∈N). Observe that each function eais an eigenvector for every Lnwith eigenvalue Φn(a). Consider the open subsets Aand Bprovided by the condition (P). Fix a compact subset K⊂G, a funcion f∈H(G) and ε > 0. Since Gis a Runge domain, a polynomial P(z) of Ncomplex variables can be found in such a way that |f(z)−P(z)|< ε/2 for all z∈K. There exists h∈HS(S=Aor B) with 17 |P(z)−h(z)|< ε/2 for all z∈K. Therefore |f(z)−h(z)|< ε for all z∈K. This shows that HAand HBare also dense subsets of H(G). It now suffices to apply part 1) of Theorem 7 on X=H(G), A={ea:a∈A},B={eb:b∈B}and Tn=Ln(n∈N). b) This part is obvious from a), together with the remark for the case N= 1 at the beginning of this section. We would have anew that HAand HBare dense in H(C), so in H(G) as well. c) By Theorem 5, Lis an operator defined on H(G). Since Φ is a nonconstant entire function, the sets A= Φ−1(|z|<1) and B= Φ−1(|z|>1) are nonempty open subsets. Now use part 2) of Theorem 7 with X=H(G), T=L= Φ(D), A={ea:a∈A}and B={eb:b∈B}. Note that, like in part a), every function ea(a∈CN) is an eigenvector for Twith eigenvalue λ(T, ea) = Φ(a). d) This is essentially Theorem 3. It has been put here for the sake of completeness. It is derived as c) (G=CNhere) by using Theorem 4. It should be noted that, if L= Φ(D), then Φ is nonconstant if and only if Lis not a scalar multiple of the identity. //// For instance, we have that there is a dense Gδsubset of entire functions fon Csuch that each entire function can be uniformly approximated on compact sets by functions of the form nf(z+n) + f(n)(z) n2(n∈N). Indeed, it suffices to consider 18 the sequence Φn(z) of the third example just before the latter theorem. Note that eaD =τafor every a∈C. In view of the result on growth of Grosse-Erdmann [16] for entire functions (see Section 1), it is natural to ask what is the minimal growth allowed for a D- hypercyclic function on a bounded domain in C. The answer for Runge domains is almost trivial and is provided in Corollary 2. We denote, as usual, by g|Sthe restriction of a function gto a subset S. COROLLARY 1. Assume that G⊂CNis a Runge domain and that Lis an operator on H(CN)that commutes with each of the translation operators τa (a∈CN), and is not a scalar multiple of the identity. Then the set M={f|G:fis entire and {(Lnf)|G}∞ 1is dense in H(G)} is dense in H(G). Proof. The assertion is evident from part d) of Theorem 8 and from the fact that H(CN) is dense in H(G). //// COROLLARY 2. If G⊂CNis a bounded Runge domain, then there exists a dense subset Min H(G)such that, for every f∈M, each derivative f(n)(n∈N0) is bounded and the orbit {f(n)}∞ 1is dense in H(G). Proof. Just apply Corollary 1 with L=D. //// 19 THEOREM 9. Suppose that Gis a Runge domain of CNand Φ,Φn(n∈N) are entire functions on CN. Denote Ln= Φn(D) (n∈N). a) Suppose that every Φnis of subexponential (exponential, resp.) type and the sequence {Φn}satisfies (Q). Then there is a dense Gδsubset of H(G)(H(CN), resp.) all of whose elements are hypercyclic functions for {Ln}. b) For N= 1 the statement of a) still holds if (Q) is changed to (Q’). Proof. We can also apply part 1) of Theorem 7. Take X=Y=H(G), Tn=Ln(n∈N), A={zp:p∈N0N},B={eb:b∈B}where the set Bis furnished by hypothesis (Q) (or by (Q’) if N= 1). Observe that span A (= {polynomials}) is dense in H(G). Each function ebis an eigenvector for every Tnwith eigenvalue λ(Tn, b) = Φn(b). Fix two finite subsets F1={zp1, ..., zpr} ⊂ A and F2={eb1, ..., ebs}⊂B. From (Q) (or (Q’)), a subsequence {nk}of positive integers can be found for the finite set F={b1, ..., bs} ⊂ Bin such a way that limk→∞ λ(Tnk, bj) = limk→∞ Φnk(bj) = ∞for all j∈ {1, ..., s}. On the other hand, if α= max{|p1|, ..., |pr|}, there is n0∈Nsuch that m(Φn)> α for all n > n0, so Φn(D)zpj= 0 for all j∈ {1, ..., r}because Dpzpj= 0 for all j∈ {1, ..., r}and for every multi-index pwith |p|> α. Consequently, each zpjis an eigenvector for Tnk (we can assume nk> n0for all k) with eigenvalue λ(Tnk, zpj) = 0, which trivially tends to zero as k→ ∞. //// 20 Unfortunately, one cannot expect any hypercyclicity result for an antidifferential operator Ψ(D−1). THEOREM 10. Assume that G⊂Cis a simply connected domain. Fix a point a∈Gand consider the corresponding antiderivative operator D−1. Suppose that Ψ and Ψn(n∈N)are in S(1/∆(a, G)) and that Ψn(z) = ∑∞ j=0 c(n) jzj. Let L, Lnbe the operators L= Ψ (D−1), Ln= Ψn(D−1) (n∈N). We have: a) If {Ln}is hypercyclic, then the sequence {c(n) 0:n∈N}is dense in C. b) Lis not hypercyclic. Proof. Land Ln(n∈N) are well defined operators by Theorem 6. If f∈H(G) is hypercyclic for {Ln}then, given b∈C, some subsequence {Lnkf}of {Lnf} must approximate the constant function g(z)≡bon the compact set {a}. But (Lnf)(a)=(∑∞ j=0 c(n) jD−jf)(a) = c(n) 0f(a) because D−jf(a) = 0 for all j∈N. This implies that c(nk) 0f(a)→b(k→ ∞), so {c(n) 0:n∈N}is dense in C. This proves a). Part b) is an unpleasant consequence of a): indeed, assume that Ψ(z) = ∑∞ j=0 cjzjand put Ln=Ln. Then c(n) 0=cn 0and for each c0∈Cthe sequence {cn 0}is not dense in C. //// Nevertheless, a sort of “pseudo-hypercyclicity” is true, as our next theorem shows. For this, let us cite the following result of W. Luh [21]: For every simply connected domain G⊂Cthere exists a sequence {Cn}∞ 1⊂Cwith the property 21 that for every φ∈H(G) the set {Qn(z) = D−nφ(z) + ∑n−1 j=0 Cn−j j!zj:n∈N}is dense in H(G). Note that the coefficients Cn’s do not depend upon φ. Just a remark before the theorem. Let G⊂Cbe a simply connected domain and fix a point a∈G. If Ψ(z) = ∑∞ j=0 cjzjis a formal power series, then Ψ ∈ S(1/∆(a, G)) if and only if α(Ψ, δ) is finite for all δ∈(0,∆(a, G)), where we have set α(Ψ, δ) = |c0|+ sup j∈N |cj|δj−1 (j−1)! . THEOREM 11. Assume that G⊂Cis a simply connected domain. Fix a point a∈Gand consider the corresponding antiderivative operator D−1. Then there exists a sequence {Cn}∞ 1⊂Csatisfying the following property: For every f∈H(G)and every sequence {Ψn(z)}∞ 1⊂S(1/∆(a, G)) of formal power series for which α(Ψn, δ)→0 (n→ ∞)for all δ∈(0,∆(a, G)), the sequence {Ψn(D−1)f(z) + ∑n−1 j=0 Cn−j j!zj:n∈N}is dense in H(G). Proof. Denote Ln= Ψn(D−1) and assume that Ψn(z) = ∑∞ j=0 c(n) jzj(n∈N). From Theorem 6 we have that every Lnis an operator on H(G). We apply the mentioned result of Luh [21] to the domain Gand the function φ= 0. We obtain that there is a sequence {Cn}∞ 1⊂Csuch that the set {Hn}∞ 1given by Hn(z) = ∑n−1 j=0 Cn−j j!zjis dense in H(G). Fix a function f∈H(G) and a compact set K⊂G. 22 If the final steps of the proof of Theorem 6 are watched then one sees that there are a compact set L⊂Gand positive constants σ, M, M1with M1∈(M, ∆(a, G)) such that ||Lnf||K≤Bn· ||f||L, where Bn=|c(n) 0|+σM1 M1−M·sup j∈N |c(n) j|Mj−1 1 (j−1)! for all n∈N. But α(Ψn, M1)→0 (n→ ∞) by hypothesis, so limn→∞ Bn= 0 and ||Lnf||K→0 (n→ ∞). Hence {Lnf}∞ 1converges uniformly to zero on compact sets. Since {Hn}∞ 1is dense in H(G), we have that {Lnf+Hn}∞ 1is also dense in H(G), as required. //// By using [21, Lemma 3] one can easily establish that for every compact set B⊂ Cwith connected complement and every function gwhich is continuous on Band holomorphic in the interior of B, there is a subsequence of {Lnf+Hn}∞ 1converging to guniformly on Band, in addition, for every Lebesgue-measurable set E⊂G and every Lebesgue measurable function g:E→C∪ {∞}, there is a subsequence of {Lnf+Hn}∞ 1converging almost everywhere to gon E. Theorem 11 together with this remark generalizes [5, Theorem 5]: in fact there we dealt with the case Ψn(z) = cnzn, where {cn}∞ 1⊂Cis a sequence such that lim supn→∞(|cn| n!)1/n ≤ 1/∆(a, G). We propose here as an open problem to give conditions on {Ψn}which guarantee the hypercyclicity of {Ψn(D−1)}. Note that this sequence can certainly be 23 hypercyclic. Indeed, for a= 0 the constant function f≡1 is {Ψn(D−1)}-universal if and only if the set {∑∞ j=0(c(n) j/j!)zj:n∈N}is dense in H(G). Appropriate c(n) jcan always be found. To finish, we establish a result about hypercyclicity of functions with the additional property that certain derivatives do not vanish on the domain. Notice that there is no D-hypercyclic entire function fsuch that f·f′·f′′ is zero-free, since {f∈H(C) : f·f′·f′′ is zero-free}={eαz+β:α, β ∈C, α = 0}(see [12, p. 433] and [24]). If q∈N0, let us denote A(q) = {f∈H(G) : f(q)(z)f(q+1)(z)= 0 for all z∈G}. Since exp ∈∩q∈N0A(q), every A(q) is nonempty. If L={Ln:n∈N}is a sequence of continuous mappings from Xinto Y, then we denote by HC(L) the set of hypercyclic elements for L. If A⊂X, denote L|A={Ln|A:n∈N}. We are now ready to state our theorem on hypercyclicity. Herzog’s result [17] is the special case q= 0, G=C,L={Dn:n∈N}while the result of the author in [6] is the special case G=C,L={Ln:n∈N}with L= Φ(D). THEOREM 12. Assume that G⊂Cis a simply connected domain and that {Φn:n∈N}is a sequence of entire functions of subexponential type satisfying (P’) or (Q’). Fix q∈N0and set A=A(q),L={Ln:n∈N}, where Lnis the operator on H(G)given by Ln= Φn(D) (n∈N). Then the set HC(L|A)is residual in A. Proof. The proof is very similar to that in [6], so we merely indicate some 24 necessary changes. The closed discs D(0, k) (k∈N) in the proof of Theorem 5 of [6] should be replaced to the closure Gkof Gk, where {Gk:k∈N}is a sequence of simply connected domains such that every Gkis compact, Gk⊂Gk+1 and G=∪∞ k=1 Gk. The metric d(f, g) in that paper is here changed to d(f, g) = ∞ ∑ j=1 1 2j ||f−g||j 1 + ||f−g||j (f, g ∈H(G)), where ||h||j= maxz∈Gj|h(z)|. From Theorems 8 and 9, HC(L) is residual in H(G) and an adequate application of Theorem 2.1 of [17] will give the result if one takes into account that A=∩∞ k=1 Akwhere Ak={f∈H(G) : minGk|f(q)·f(q+1)|>0}. If k∈Nis fixed, then there is a simply connected subdomain U⊂Gsuch that Gk⊂Uand f(q)(z)f(q+1)(z)= 0 for all z∈U. The existence of an approximating sequence {Pm}of polynomials on D(0, k +ε) in [6, Theorem 5] is here guaranteed by Runge’s theorem, which should be applied on V,Vbeing a simply connected domain such that Vis compact and Gk⊂V⊂V⊂U. We left the details to the reader, which should find no difficulty if he follows step by step the proof of the cited reference. //// ACKNOWLEDGEMENT The author thanks the referees for some valuable comments and suggestions. 25