On diagonal operators between the sequence (LF)-spaces

Diagonal (multiplication) operators acting between a particular class of countable inductive spectra of Fréchet sequence spaces, called sequence (LF)-spaces, are investigated. We prove results concerning boundedness, compactness, power boundedness, and mean ergodicity. Furthermore, we determine when a diagonal operator is Montel and reﬂexive. We analyze the spectra in terms of the system of weights deﬁning the spaces.


Introduction
In this paper, we are concerned with studying diagonal (multiplication) operators acting on a particular class of sequence (LF)-spaces. Many authors have investigated these mappings, for instance, in the setting of weighted spaces of (vector-valued) continuous functions by Manhas [17,18] and Oubbi [20] among others. Recently many authors have focused on studying the ergodic properties of the diagonal operators acting on (LB)-spaces of functions and sequences, for instance, [10,22]. In the context of Köthe echelon spaces, Crofts [12] investigated diagonal operators. The case of multiplication operators on weighted spaces of analytic functions on the complex unit disc was studied by Bonet and Ricker [11]. Albanese and the author in [5] have studied the spectra and the ergodic properties of the multiplication operators on the space S(R) of rapidly decreasing functions and the (PLB)-space of its multipliers. Echelon and co-echelon spaces were studied by Köthe and Toeplitz. In a paper [24] published in 1992, Vogt characterized the regularity, completeness, and (weak) acyclicity of Köthe (LF)-sequence spaces E p , 1 ≤ p ≤ ∞ ∪ {0}. • regular, if every bounded subset of E is contained and bounded in some step E n ; • (pre)compactly retractive, if for every (pre)compact subset K of E, there exists m ∈ N such that K ⊂ E m and it is (pre)compact there; • strongly boundedly retractive, if it is regular and for all k ∈ N, there exists l ≥ k such that (E, t) and (E l , t l ) induce the same topology on each bounded set of (E k , t k ); • boundedly retractive, if every bounded subset B of E is contained in some step E n and the topologies of E and E n coincide on B; • sequentially retractive, if every convergent sequence in E is contained in some step E n and converges there.
It is well-known that every complete (LF)-space is regular, but whether the converse holds seems to be an open problem (mentioned by Grothendieck), even for (LB)-spaces. We refer the reader to [6,24,25] for more details. The space (E, t) is said to satisfy the condition (M) (resp. (M 0 )) of Retakh [21] if there exists an increasing sequence (U n ) n∈N of subsets of E such that for all n ∈ N U n is an absolutely convex 0-neighborhood of E n such that ∀n ∈ N ∃m ≥ n ∀μ ≥ m : t μ and t m induce the same topology on U n .
The following important theorem gives some equivalence of the concepts mentioned above. This theorem is due to Wengenroth for (LF)-spaces. See [25,Theorem 6.4].

Theorem 2.1
For an (LF)-space E = ind n E n the following conditions are equivalent: (1) There is an increasing sequence (U n ) n∈N of subsets of E such that for all n ∈ N U n is an absolutely convex 0-neighborhood of E n for which for all n ∈ N there exists m ≥ n such that t and t m induce the same topology on U n ; Furthermore, the condition (M) implies the completeness of the (LF)-spaces (see [25,Corollary 6.5]). Therefore, from the above considerations, we have that if an (LF)-space E = ind n E n satisfies the condition (M), then E is strongly boundedly retractive.
Valdivia in [24,Page 161] showed that an (LF)-space E = ind n E n satisfies the condition (M 0 ) if, and only if, for all m ∈ N there is an absolutely convex 0-neighborhood U m of E m with U m ⊆ U m+1 such that, given any n ∈ N there is an integer μ > n such that σ (E, E ) and σ (E μ , E μ ) coincide on U m .

Operators acting on (LF)-spaces
A linear operator between the lcHs X and Y is called bounded if it maps some 0-neighborhood of X into a bounded subset of Y , while it is said to be compact if it maps some 0-neighborhood of X into a relatively compact subset of Y . In the following, we characterize the boundedness and compactness of operators acting between (LF)-spaces. For this issue, we denote by B(X ) the set of the bounded subsets of a lcHs X .
We start recalling a known result of Grothendieck [14] and a similar one.

Lemma 2.2 (1) Let G be a metrizable lcHs. Then for every family of bounded subsets
(2) Let G be a metrizable lcHs. Then for every family of precompact subets (C j ) j∈N of G, there exists a sequence (λ j ) j∈N ∈ (0, ∞) N such that ∞ j=1 λ j C j is precompact in G. We use the previous lemma to give the following characterizations.

Proposition 2.3
Let E = ind n E n and F = ind n F n be two (LF)-spaces. The following assertions hold: (1) Assume that F is regular. Then the linear operator T : E → F is bounded if, and only if, there exists n ∈ N such that for all m ∈ N we have that T (E m ) ⊂ F n and the restriction T : , which is a precompact subset of F n , since the absolutely convex hull of a precompact set in a lcHs is still precompact. Therefore, we obtain that T (U ) is precompact in F n and so in F.
Given X , Y two lcHs, a linear operator T : X → Y is called Montel if it maps bounded subsets of X into relatively compact subsets of Y . If X and Y are Banach spaces, then T : X → Y is Montel if, and only if, it is compact. For an operator between (LF)-spaces we have the following result. Proof Suppose that T : E → F is Montel. Fixed m ∈ N, by the continuity of T there exists n ∈ N such that T : E m → F n is continuous (see [14]). Since F is in particular strongly boundedly retractive, we choose a n ≥ n as in the definition of the condition and prove that T : E m → F n is Montel. If we take B a bounded subset of E m , due to the continuity of T we have that T (B) is bounded in F n . Moreover, by assumption, T (B) is relatively compact in F. Hence, the topologies on T (B) induced by F and F n coincide, since F is strongly boundedly retractive. This means that T (B) is relatively compact also in F n .
We assume now that the condition is fulfilled and prove that T : E → F is Montel. Fix a bounded subset B of E. Due to the regularity of E, we can find m ∈ N such that B is bounded in E m . By assumption, there exists n ∈ N such that the restriction T : E m → F n is Montel. Hence, T (B) is relatively compact in F n and so in F.

Remark 2.5
In the proof of Proposition 2.4, to show that the condition is sufficient, we only use the assumption of the regularity of E.
We recall that given X , Y two lcHs, an operator T : X → Y is called reflexive if it maps bounded subsets of X into relatively weakly compact subsets of Y . If Y is reflexive, then a continuous linear operator T : X → Y is reflexive. We refer the reader to [16] for more details.
We give the following characterization concerning (LF)-spaces. Proof Suppose that T : E → F is reflexive. Fixed m ∈ N, by the continuity of T there exists n ∈ N such that T : E m → F n is continuous (see [14]). If we take B a bounded subset of E m , due to the continuity of T we have that T (B) is bounded in F n . Since F satisfies the condition (M 0 ), taking into account Valdivia's result [24,Page 161] there exists an increasing sequence (U k ) k∈N of subsets of F such that U k is an absolutely convex 0-neighborhood of F k for all k ∈ N and the topologies induced on U k from σ (F, F ) and σ (F n , F n ) coincide, for some n > k. From the boundedness of T (B), we can find λ > 0 such that T (B) ⊂ λU n . Moreover, by assumption, T (B) is relatively weakly compact in F. This implies that T (B) is relatively weakly compact in F n , that means that T : E m → F n is reflexive.
We assume now that the condition is fulfilled and prove that T : is relatively weakly compact in F n and so in F.

The sequence (LF)-spaces l p (V)
In this section, we introduce the sequence (LF)-spaces l p (V) and recall the main properties of these spaces concerning regularity and completeness.
For all n ∈ N, V n = v n,k k∈N is a countable family of (strictly) positive sequences, called weights, on N. We denote by V the sequence (V n ) n∈N and we assume that the following two conditions are satisfied: Given a system of weights V as above, for n, k ∈ N and 1 ≤ p ≤ ∞ we define as usual where · p denotes the usual l p norm. For p = 0 we set These spaces are Banach with the corresponding p v n,k norms and c 0 (v n,k ) is Banach with the norm inherited from l ∞ (v n,k ). Since l p (v n,k+1 ) is continuously embedded into l p (v n,k ), the sequence {l p (v n,k )} k∈N of Banach spaces forms a projective spectrum. Hence, for all n ∈ N and 1 ≤ p ≤ ∞, we can consider the echelon spaces Endowed with the projective topologies λ p (V n ) = proj k l p (v n,k ) (resp. λ 0 (V n ) = proj k c 0 (v n,k )), these spaces are Fréchet with the topology defined by the corresponding seminorms p n,k := p v n,k , k ∈ N.
of Fréchet spaces forms an inductive spectrum and the spaces To describe the inductive topology of these spaces, we can associate to l p (V) the Nachbin family in the usual way (see [8,24]). Set for some α n > 0, k(n) ∈ N and for all i ∈ N. For 1 ≤ p ≤ ∞ we define in the usual way the Banach spaces The spaces K p (V ), for 1 ≤ p ≤ ∞ ∪ {0}, are complete, being a projective limit of complete spaces, and their seminorms will be denoted by p v , v ∈ V , when p is fixed and no confusion shall arise.
, is a topological isomorphism into as a consequence of [24, Proposition 5.1]. For p = ∞, the inclusion l ∞ (V) → K ∞ (V ) also holds. We need to require, in addition, that the system of weights V satisfies the following condition (see [8,Theorem 7]): where W m denotes the system of all non-negative sequences which are dominated by In the (LB)-case, condition (3.1) is equivalent to condition (D) of Bierstedt and Meise (see [7]).
Furthermore, the inclusion In the following, we recall the conditions for which we have that l p (V) = K p (V ) algebraically and also topologically. Due to Remark 3.1, In our context, the characterization of the regularity is due to Vogt [24], in terms of a condition on the system of weights V. Definition 3. 2 We say that the sequence V = v n,k k∈N n∈N satisfies the condition (WQ) (or is of type (WQ)) if

Definition 3.3 We say that the sequence
In [24], is shown the link between the (M) and (M 0 )-conditions (i.e., the acyclicity and weak-acyclicity) and the (Q) and (WQ)-conditions. We find the characterization of the regularity of the (LF)-spaces l p (V) in the following theorem of Vogt (see [24]).

Diagonal operators acting on l p (V)
From now on, we work with diagonal (multiplication) operators on the sequence (LF)-spaces l p (V). We denote by ω the space of all the sequences ϕ = (ϕ i ) i∈N ∈ C N . Given a sequence ϕ = (ϕ i ) i∈N ∈ ω, we define the multiplication (or diagonal) operator as M ϕ : ω → ω such that (x i ) i∈N → (x i ϕ i ) i∈N . If M ϕ acts continuously from a sequence lcHs X into a sequence lcHs Y , we say that ϕ is a multiplier from X to Y . Firstly, we characterize the multipliers between the sequence (LF)-spaces l p (V).
(3) For all m ∈ N there exists n ∈ N such that for all k ∈ N there exist l ∈ N for which Proof Clearly, (2) implies (1) and (1) . This proves that the graph of M ϕ is closed. We prove the statement for 1 ≤ p ≤ ∞. The proof in the case p = 0 is analogous and so is omitted.  A similar characterization holds for the boundedness of multiplication operators between the sequence (LF)-spaces l p (V). We need to recall the characterization of boundedness for operators acting between Fréchet spaces (see [4,Lemma 25]    The next target is to describe when a diagonal operator between the sequence (LF)-spaces l p (V) is compact. Before giving the result, we need to recall the characterization of the compactness of multiplication operators between the Banach spaces l p and c 0 . The proof is left to the reader.

Lemma 4.5
Let φ ∈ ω. The following assertions hold: As for the boundedness, we also need a characterization of the compactness for operators acting between Fréchet spaces. Again we refer to [4].
Thus, M ϕ is compact if, and only if, M φ : l p → l p is compact. Due to Lemma 4.5, this holds if, and only if, φ ∈ c 0 , i.e., if φ vanishes at infinity.
The same also holds for p = 0.
To describe when diagonal operators between the sequence (LF)-spaces l p (V) are Montel, firstly, we characterize when diagonal operators between the echelon spaces are Montel.

Remark 4.8
We recall a known result about the relative compactness for l p (see [19,Chapter 15]). For 1 ≤ p < ∞, a subset K of l p is relatively compact if, and only if, for every ε > 0 there exists j 0 ∈ N such that for every x ∈ K we have ∞ j= j 0 +1 |x j | p < ε p . Analogously, for p = 0 a subset K of c 0 is relatively compact if, and only if, for every ε > 0 there exists j 0 ∈ N such that for every x ∈ K we have sup j≥ j 0 +1 |x j | < ε.
Let A = (a n ) n∈N denote a Köthe matrix, i.e., an increasing sequence of strictly positive functions a n . We consider λ ∞ (A) + = {x = (x i ) i∈N ∈ ω | (a n x) n∈N ∞ < ∞ and x i > 0 ∀i ∈ N}.
The following useful description of the bounded sets in a Köthe echelon space is due to Bierstedt et al. [9].

Proposition 4.9 Let A = (a n ) n∈N be a Köthe matrix. For 1 ≤ p < ∞ ∪ {0}, a subset B of λ p (A) is bounded if, and only if, there exists a ∈ λ ∞ (A)
First of all, we study the case p < ∞. Fixed a weight v, we recall that c 0 is isomorphic . Hence, taking into account Remark 4.8, a subset K of c 0 (v) is relatively compact if, and only if, for every ε > 0 there exists j 0 ∈ N such that for every x ∈ K we have sup j≥ j 0 +1 v( j)|x j | < ε. Analogously for 1 ≤ p < ∞. We assume now that the condition is fulfilled and prove that M ϕ : λ 0 (A) → λ 0 (B) is Montel. We want to prove that for a fixed bounded subset B of λ 0 (A) the set M ϕ (B) is relatively compact in c 0 (b m ) for all m ∈ N. To do this, since B is bounded, we apply Proposition 4.9 and choose a ∈ λ ∞ (A) + such that B ⊂ B a . It suffices to show that M ϕ (B a ) is relatively compact in c 0 (b m ) for all m ∈ N, that is, using Remark 4.8, for every ε > 0 and m ∈ N there exists j 0 ∈ N such that for every y ∈ M ϕ (B a ) we have sup j≥ j 0 +1 b m ( j)|y j | < ε. Hence, fixed m ∈ N and ε > 0, since (4.4) holds, we can choose j 0 ∈ N such that a( j)|ϕ j |b m ( j) < ε for all j ≥ j 0 + 1. If y ∈ M ϕ (B a ), then y = (y i ) i∈N = (ϕ i x i ) i∈N and |x i | ≤ a(i) for all i ∈ N. Then, if j ≥ j 0 + 1 and y ∈ M ϕ (B a ), we get Therefore, the case p < ∞ is characterized. Now we prove that the same characterization holds for p = ∞. We suppose that M ϕ : λ 0 (A) → λ 0 (B) is Montel. Applying [13,Corollary 2.3], then M t ϕ : λ 0 (A) b → λ 0 (B) b is Montel. Note that λ 0 (A) b and λ 0 (B) b are complete (LB)-spaces (see [6,Proposition 10]). Then applying [13,Corollary 2.4], we get that M tt ϕ : is the bidual of λ 0 (B)), we get the claim.
Thus, we can characterize when a diagonal operator between the sequence (LF)-spaces l p (V) is Montel. The proof is an application of Propositions 2.4, 4.10 and 4.11. To study the Köthe echelon case, we have to make a distinction. For p = 1, 0, ∞, we want to show that a diagonal operator acting between the echelon spaces is Montel if, and only if, it is reflexive. Let A = (a n ) n∈N , B = (b m ) m∈N be two Köthe matrices and fix ϕ = (ϕ i ) i∈N ∈ ω.  But in general, the diagonal operator does not necessarily have to be Montel, since it is not Montel even between the Banach spaces l p (take a n = b n = 1). In this case, what we have is this characterization. The proof is an obvious consequence of Theorem 3.4. Proposition 4.14 Let V, W be two systems of weights and fix ϕ = (ϕ i ) i∈N ∈ ω. For 1 < p < ∞, assume that l p (W) is regular. Then the diagonal operator M ϕ : l p (V) → l p (W) is continuous if, and only if, it is reflexive.

Spectrum of diagonal operators acting on l p (V)
Let X be a lcHs and let L(X ) denote the space of all continuous linear operators from X into itself. Given T ∈ L(X ), the resolvent set of T is defined by Unlike for Banach spaces, it may happens that ρ(T , X ) = ∅ or that ρ(T , X ) is not open in C (see, e.g., [3]). This is the reason why many authors consider the subset ρ * (T , X ) of ρ(T , X ) consisting of all λ ∈ C for which there exists δ > 0 such that B δ (λ) := {μ ∈ C | |μ − λ| < δ} ⊆ ρ(T , X ) and the set {(μI − T ) −1 | μ ∈ B δ (λ)} is equicontinuous in L(X ). If X is a Fréchet space, then it suffices that this set is bounded in L s (X ), where L s (X ) denotes L(X ) endowed with the strong operator topology. The advantage of ρ * (T , X ), whenever it is not empty, is that it is open and the resolvent map is holomorphic from ρ * (T , X ) into L b (X ) (see, e.g., [2,Proposition 3.4]), where L b (X ) denotes L(X ) endowed with the topology of the uniform convergence on bounded subsets of X ). Define the Waelbrock spectrum σ * (T , X ) := C \ ρ * (T , X ), which is a closed set containing σ (T , X ).
We start studying the point spectrum of the diagonal operators. The proof of this result is standard and so is omitted (see [4] for an analogous one).

Lemma 5.1 Let V be a system of weights and fix
We determine the resolvent set of the diagonal operators acting between the sequence (LF)-spaces l p (V). Compare with [22, Proposition 1] for Köthe echelon spaces. Proposition 5.2 Let V be a system of weights and fix ϕ = (ϕ i ) i∈N ∈ ω. For 1 ≤ p ≤ ∞ ∪{0}, the following assertions are equivalent: (1) μ ∈ ρ(M ϕ , l p (V)); (2) For all m ∈ N there exists n ∈ N such that for all k ∈ N there exist l ∈ N for which sup i∈N v n,k (i) v m,l (i)|ϕ i − μ| < ∞. (5.1)