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Invariant subspaces of the integration operators on H\"ormander algebras and Korenblum type spaces

We describe the proper closed invariant subspaces of the integration operator when it acts continuously on countable intersections and countable unions of weighted Banach spaces of holomorphic functions on the unit disc or the complex plane. Applications are given to the case of Korenblum type spaces and H\"ormander algebras of entire functions.


Introduction.
Let G be the open unit disc D or the whole complex plane C. We denote by H(G) the Fréchet space of holomorphic functions on G, endowed with the topology of uniform convergence on compact subsets of G. A space E of holomorphic functions on the domain G is a Hausdorff locally convex space that is a subset of H(G), such that the inclusion map E ⊂ H(G) is continuous and E contains the polynomials. If E is a Fréchet space or a countable inductive limit of Banach spaces, by the closed graph theorem, the inclusion map E ⊂ H(G) is continuous if and only if the point evaluations E → C, f → f (z) at all the points z ∈ G are continuous on E. We are mainly interested in the case when the polynomials are dense in E. In this case then E is separable.
Banach spaces of holomorphic functions on the unit disc D and on the complex plane C have been thoroughly investigated. We refer the reader for example to the books [23], [34] and [35]. Hörmander algebras of entire functions [1], [10], [11], [17], [28], as well as Koremblum space and other intersections and unions of growth Banach spaces of holomorphic functions on the unit disc [6], [18], [23], are natural examples of (locally convex) spaces of holomorphic functions. Vogt [32] proved that there are Fréchet spaces E which are contained in H(G) such that the inclusion E ⊂ H(G) is not continuous.
In this note we study the set of proper closed invariant subspaces of the integration operator which describe the invariant subspaces when the integration operator acts on Korenblum type spaces, and Theorem 5.1, which explains the situation in case of some Hörmander algebras of entire functions. The proofs of these results depend on some abstract Theorems 3.2 and 3.3 and they rely heavily on Theorems 3.8 and 3.16 due to Abanin and Tien [4]. A different method permits us to handle the (LB)-algebra of entire functions of exponential type in Theorem 5.
3. An open question about the invariant subspaces of the integration operator on certain Hörmander algebras is mentioned in Remark 5.4.
Abanin and Tien describe in [4] the closed invariant subspaces of the integration operator on various scales of weighted Banach spaces of holomorphic functions. As mentioned above, some of their results are very important for our theorems below. We refer the reader to the introduction of [4] for classical results about invariant subspaces of the integration operators and more recent ones in [7], [19] and [20]. The continuity of the integration operator on weighted Banach spaces of holomorphic functions was investigated by Harutyunyan and Lusky [22]; see also [2] and [5]. Other aspects, like spectrum and ergodic or dynamical properties, were considered by Beltrán, Fernández and the first author in [8]. Similar questions for operators defined on Hörmander algebras were investigated in [9].
Our notation for functional analysis, in particular for locally convex spaces, Fréchet spaces and (LB)-spaces, is standard. We refer the reader to [12], [24], [29] and [31]. If E is a locally convex space, its topological dual is denoted by E ′ . The linear span of a subset A of E is denoted by span(A). The closure of a subset A in E is denoted by A, and A in case the space in which the closure is taken must be emphasized.
2 Results about subspaces of Fréchet or (LB)-spaces. Lemma 2.1 Let X := proj n X n be a Fréchet space such that X = n∈N X n with each (X n , · n ) a Banach space. Moreover, it is assumed that X is dense in X n and that X n+1 ⊆ X n with a continuous inclusion for each n ∈ N. Let M be a subspace of X. Then (ii) If M is proper and closed, then there is n(0) ∈ N such that M Xn is proper in X n for each n ≥ n(0).
Proof. (i) Since the inclusions X ⊂ X n+1 ⊂ X n are continuous for each n ∈ N, we clearly have we have x ∈ n≥1 X n = X. Moreover, for each n ∈ N there is y(n) ∈ M such that x − y(n) n < 1/n. Then x = lim n→∞ y(n) in X and x ∈ M X .
(ii) First of all, since {0} = M , we get {0} = M Xn for each n ∈ N. Proceeding by contradiction, suppose that there is an increasing sequence (n(k)) k of natural numbers such that M X n(k) = X n(k) for each k ∈ N. Since X := proj k X n(k) , we could apply part (i) to conclude M X = X. Since M is closed by assumption, we have M = X, and M would not be a proper subspace. ✷ Lemma 2.2 Let X = ind n X n be an (LB)-space with increasing union of Banach spaces X = n∈N X n . If M is a proper closed subspace of X, then there is n(0) ∈ N such that M ∩ X n is a proper closed subspace of X n for each n ≥ n(0).
Proof. Since the inclusion X n ⊂ X is continuous for each n ∈ N, the set M ∩ X n is closed in X n for each n ∈ N. As M is proper, there is x ∈ M, x = 0. There is n(1) ∈ N such that x ∈ M ∩ X n (1) . On the other hand, since M = X, there is y ∈ X \ M . Select n(0) ≥ n(1) such that y ∈ X n(0) . Then, if n ≥ n(0), we have x ∈ M ∩ X n and y ∈ X n \ M . Thus, M ∩ X n is a proper subspace of X n ✷ 3 Abstract results about invariant subspaces of the integration operator on spaces of holomorphic functions.
Our first Lemma is stated in page 412 of [4]. We include a proof for the sake of completeness, because it is very relevant in our considerations below. As in [4], given a space of holomorphic functions E on the open unit disc D or the complex plane C, and N ∈ N, we set Since the polynomials are dense in E by assumption, there is a sequence of polynomials (g k ) k such that lim k→∞ g k = f in E. This implies that lim k→∞ g k = f in H(G). Therefore, lim k→∞ Theorem 3.2 Let F := proj n F n be a Fréchet space such that F = n∈N F n with each F n a Banach space of holomorphic functions on the open domain G = D or G = C, such that the polynomials are contained in F and dense in F n for each n ∈ N. Assume that the integration operator J : F n → F n is continuous for each n ∈ N, and that, for each n ∈ N, every proper closed invariant subspace of J on F n is of the form A Fn K for some K ∈ N. Then (i) J : F → F is continuous, and (ii) Every proper closed invariant subspace for J on F is of the form Proof. (i) The continuity of J : F n → F n for each n ∈ N clearly implies J(F ) ⊂ F and that J : F → F is continuous, because F is the projective limit of the sequence of Banach spaces (F n ) n .
(ii) It is easy to see that A F K is a closed invariant subspace of the integration operator J on F ; see for example Lemma 3.1.
Now, let M be a proper closed subspace of F which is invariant for the integration operator J. By Lemma 2.1 (ii) there is n(0) ∈ N such that M Fn is proper (and closed) in By assumption, for each n ≥ n(0), there is k(n) ∈ N such that Take, for each n ∈ N, k(n) as the minimum of those j ′ s such that z j ∈ M Fn . Clearly We claim that the sequence (k(n)) n of natural numbers must be bounded. If this is not the case, lim n→∞ k(n) = ∞. For each n ≥ n(0) we have Therefore, the Taylor coefficients (a j (f )) ∞ j=0 of each f ∈ M must vanish; that is, M = {0}, which is a contradiction, as M is proper.
As a consequence of the claim that we have just proved, there are K ∈ N and n(1) ≥ n(0) such that k(n) = K for each n ≥ n(1). This implies for each n ≥ n(1). Thus, we may apply Lemma 2.1 (i) to get This completes the proof. ✷ Assume that the integration operator J : E n → E n is continuous for each n ∈ N, and that, for each n ∈ N, every proper closed invariant subspace of J on E n is of the form A En K for some K ∈ N. Then (i) J : E → E is continuous, and (ii) Every proper closed invariant subspace of J on E is of the form Proof. (i) Clearly J(E) ⊂ E and J : E → E is continuous since E = ind n E n and J is stepwise continuous by assumption.
(ii) The assumptions imply that the polynomials are dense in E. The subspaces A E K are proper closed subspaces of E which are invariant for the integration operator J.
We fix a proper closed subspace M of E which is invariant for J. By Lemma 2.2 there is n(0) ∈ N such that M ∩ E n is a proper closed subspace of E n for each n ≥ n(0). It is also invariant for J, since J(M ) ⊂ M and J(E n ) ⊂ E n . By our assumptions, for each n ≥ n(0) there is k(n) ∈ N such that We select, for each n ∈ N, k(n) as the minimum of those j ′ s such that z j ∈ M ∩ E n . Since M ∩ E n ⊂ M ∩ E n+1 for each n ∈ N, we have 1 ≤ k(n + 1) ≤ k(n) ≤ k(n(0)) for each n ≥ n(0). Then there are L ∈ N and n(1) ≥ n(0) such that k(n) = L for each n ≥ n(1). This yields, for each n ≥ n(1), A systematic study of inductive and projective limits of weighted Banach spaces of type H ∞ v and H 0 v was initiated by Bierstedt, Meise and Summers in [15]. If V = (v n ) n is a decreasing sequence of weights on G = D or G = C, we define the weighted inductive limit by V 0 H(G) := ind n H 0 vn (G). These (LB)-spaces have been investigated by many authors; see e.g. [5], [13] and [14] and the references therein. On the other hand, if A = (a n ) n is an increasing sequence of weights on G = D or G = C, the weighted Fréchet space of holomorphic functions on G is defined by A 0 H(G) := proj n H 0 an (G). Fréchet spaces of this type were studied by E. Wolf [33]. Concrete examples of spaces of this type in the case of entire functions appear in Section 5. The most relevant examples in the case of holomorphic functions on the disc are Koremblum type spaces, which we define now.
For each µ > 0, the growth Banach spaces of holomorphic functions are defined as A −µ := H ∞ vµ and A −µ 0 := H 0 vµ for the standard weight v µ (r) = (1 − r) µ . These Banach spaces play a relevant role in connection with interpolation and sampling of holomorphic functions; see [23,Section 4.3]. The spaces of holomorphic functions of Korenblum type are defined as follows.
in which case also The space A −γ + is a Fréchet space when endowed with the locally convex topology generated by the increasing sequence of norms |||f ||| k := sup z∈D (1 − |z|) γ+ 1 k |f (z)|, for f ∈ A −γ + and each k ∈ N. Clearly, A −γ + is a Fréchet space of holomorphic functions of type A 0 H(D) for an increasing sequence A = (a n ) n .
Each space A −γ − is endowed with the finest locally convex topology such that all the natural inclusion maps A −µ ⊂ A −γ − , for µ < γ, are continuous. In particular, is an (LB)-space. Of course, the inductive limit is taken over all k ∈ N such that (γ − 1 k ) > 0. The Korenblum space (see [25])  Proof. For each weight v on the unit disc D, it follows from [4, proposition 3.1 and Theorem 3.8] (see also [2]), that the integration operator J :

is continuous (even compact), and that every proper closed invariant subspace of
(1) The integration operator on the Fréchet space A 0 H(D) = proj n H 0 an (D) satisfies all the hypothesis of Theorem 3.2. The conclusion follows from this result.
(2) Similarly, J : V 0 H(D) → V 0 H(D) satisfies the assumptions of Theorem 3.3, which permits to complete the proof. ✷ We have the following consequence of Theorem 4.1.

Corollary 4.2 The integration operator J is continuous on the Korenblum type spaces
5 Invariant subspaces of the integration operator on Hörmander algebras. endowed with the projective limit topology, for which it is a Fréchet space. These spaces are topological algebras. They are called Hörmander algebras of entire functions. Clearly A 0 p (C) ⊂ A p (C). Condition (w3) implies that A 0 p (C) contains the polynomials, and condition (w4) implies that the spaces are stable under differentiation. Weighted algebras of entire functions of this type have been considered by many authors. See e.g. [1], [10], [11], [27] and [28] and the references therein.
Here are some examples: When p(z) = |z| s , s > 0, then A p (C) consists of all entire functions of order s and finite type or order less than s; and A 0 p (C) is the space of all entire functions of order at most s and type 0. For s = 1, A p (C) is the space of all entire functions of exponential type Exp(C), and A 0 p (C) is the space of entire functions of infraexponential type.
It was proved by Beltrán, Fernández and the first author in [9, Lemma 4.1] that the integration operator J is continuous on E = A p (C) and E = A 0 p (C) for every weight exponent p. Concerning invariant subspaces we have the following result.
Proof. Define, for β > 0, the weight v β (z) := exp(−βp(z)). The polynomials are dense in H 0 v β and, by [3,Theorem 3.8], the operator J : . We recall that the norm in the Hardy space H 2 is given by In what follows we denote g ε (z) := g(εz).

Lemma 5.2
Any transcendent function f ∈ H(C) is a cyclic vector for B.
Proof. According to [21,Theorem 2.2.4], the restriction of any transcendent function to the unit circle is a cyclic vector for the backward shift on the Hardy space H 2 . We now fix a compact set K ⊂ C and g ∈ H(C). Take r > 0 so that |z| ≤ r for all z ∈ K and put R = 2r. Since where the closure is taken in H 2 . So, for every ε > 0 we can find coefficients a 1 , . . . a N such that For every z ∈ K we have We consider the topological isomorphism (see for instance [31, p.94 We identify f ∈ H(C) with T f ∈ H(C) ′′ , T f (u) := u(f ). Then For f ∈ H(C), f (z) = ∞ k=0 a k z k and g ∈ Exp (C), g(z) = ∞ k=0 b k z k , we have Identity (5.1) is proved. Consequently is a proper closed subspace of H(C) and satisfies B(F ) ⊂ F. According to Lemma 5.2, F cannot contain transcendent functions. Since F is a closed subspace of the Fréchet space H(C) and it consists only of polynomials then an application of Baire's theorem permits us to conclude that F is finite dimensional. Let f 0 (z) = N −1 k=0 a k z k + z N be an element of Finally M = M •• = g ∈ Exp (C)| Φ t (f ), g = 0 ∀f ∈ F = g ∈ Exp (C)| z k , Φ(g) = 0, 0 ≤ k ≤ N = g ∈ Exp (C)| g (k) (0) = 0, 0 ≤ k ≤ N .

Remark 5.4
The integration operator is continuous on all the Hörmander algebras defined above A 0 p (C) and A p (C) by [9,Lemma 4.1]. However, we do not know whether the conclusion of Theorem 5.1 holds if the condition lim r→∞ p ′ (r) = ∞ fails, in particular for the weight exponents p(z) = |z| s , 0 < s ≤ 1, except in the case covered by Theorem 5.3.