On diagonal operators between the sequence (LF)-spaces $l_p({\mathcal {V}})$

Abstract

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.

1 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 ${{\mathcal {S}}}({{\mathbb {R}}})$ 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 $E_p$, $1 \le p\le \infty \,\cup \{0\}$.

In this note, we treat different aspects of diagonal operators acting between the sequence (LF)-spaces $l_p({\mathcal {V}})$, extending the recent results of Rodríguez–Arenas obtained in [22] for Köthe echelon spaces. The sequence (LF)-spaces $l_p({\mathcal {V}})$ are defined as an inductive limit of a spectrum of echelon spaces. Vogt in [24] has obtained important results concerning regularity and completeness of sequence (LF)-spaces. We focus our attention on the action of the diagonal operator on them. The purpose of this work is to characterize in terms of the weight sequences the boundedness, compactness, being Montel, and reflexivity of the diagonal operator $M_\varphi :l_p({\mathcal {V}})\rightarrow l_p({\mathcal {W}}), (x_i)_{i\in {{\mathbb {N}}}}\mapsto (x_i\varphi _i)_{i\in {{\mathbb {N}}}}$, with $1\le p\le \infty \,\cup \,\{0\}$, ${\mathcal {V}},{\mathcal {W}}$ two systems of weights and $\varphi \in \omega$. Furthermore, we determine the spectra of the diagonal operators and study their ergodic properties. Our notation for functional analysis is standard. We refer the reader to [6, 16, 19].

The article is divided into six sections. In Sect. 2, we establish the notation and recall some of the most fundamental definitions concerning (LF)-spaces. Moreover, we characterize the boundedness, compactness, being Montel, and reflexivity for operators acting between (LF)-spaces. Section 3 is devoted to the definitions and the main properties of the sequence (LF)-spaces $l_p({\mathcal {V}})$. While in Sect. 4 we study the diagonal operators acting between the spaces $l_p({\mathcal {V}})$. We show when a diagonal operator $M_\varphi :l_p({\mathcal {V}})\rightarrow l_p({\mathcal {W}})$ is continuous, bounded, compact, Montel, and reflexive in terms of the weight sequences that form the countable inductive spectra of Fréchet sequence spaces. In Sect. 5, we analyze the spectrum and the Waelbroeck spectrum of the diagonal operators. A complete discussion concerning power boundedness and (uniform) mean ergodicity for diagonal operators is given in Sect. 6.

2 Definitions and results on (LF)-spaces

The purpose of this section is to recall some definitions and fundamental results of the theory of the (LF)-spaces.

2.1 (LF)-spaces

An (LF)-space is a locally convex Hausdorff space (lcHs, briefly) E which is an inductive limit $E=\mathop {\mathrm{ind\,}}_n E_n$ of an inductive sequence $(E_n)_{n\in {{\mathbb {N}}}}$ of Fréchet spaces, i.e., $E=\cup _n E_n$ and $E_n\hookrightarrow E_{n+1}$ continuously for all $n\in {{\mathbb {N}}}$ (see [14] for more details). In the following, we denote by t the lc-topology of E and by $t_n$ the Fréchet topology of each $E_n$, $n\in {{\mathbb {N}}}$. The lc-topology of the (LF)-space $E=\mathop {\mathrm{ind\,}}_n E_n$ is the finest lc-topology that makes the inclusions $E_n\hookrightarrow E$ continuous for all $n\in {{\mathbb {N}}}$.

Let $E=\mathop {\mathrm{ind\,}}_n E_n$ be an (LF)-space. E is called:

• 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 $m\in {{\mathbb {N}}}$ such that $K\subset E_m$ and it is (pre)compact there;

• strongly boundedly retractive, if it is regular and for all $k\in {{\mathbb {N}}}$, there exists $l\ge k$ 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 $(U_n)_{n\in {{\mathbb {N}}}}$ of subsets of E such that for all $n\in {{\mathbb {N}}}$ $U_n$ is an absolutely convex 0-neighborhood of $E_n$ such that

$$\begin{aligned}&\forall n\in {{\mathbb {N}}}\, \exists m\ge n\, \forall \mu \ge m: t_\mu \, \text {and}\, t_m \, \text {induce the same topology on}\, U_n. \\&(\text {resp.}\,\forall n\in {{\mathbb {N}}}\, \exists m\ge n\,\, \forall \mu \ge m: \sigma (E_\mu ,E_\mu ')\, \text {and}\, \sigma (E_m,E_m') \, \text {induce the same topology on}\, U_n ). \end{aligned}$$

(LF)-spaces with condition (M) (resp. (M$_0$)) are called acyclic (resp. weak-acyclic).

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 $E=\mathop {\mathrm{ind\,}}_nE_n$ the following conditions are equivalent:

• (1) There is an increasing sequence $(U_n)_{n\in {{\mathbb {N}}}}$ of subsets of E such that for all $n\in {{\mathbb {N}}}$ $U_n$ is an absolutely convex 0-neighborhood of $E_n$ for which for all $n\in {{\mathbb {N}}}$ there exists $m\ge n$ such that t and $t_m$ induce the same topology on $U_n$;

• (2) E satisfies the condition (M);

• (3) E is boundedly retractive;

• (4) E is (pre)compactly retractive;

• (5) E is sequentially retractive.

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 $E=\mathop {\mathrm{ind\,}}_nE_n$ satisfies the condition (M), then E is strongly boundedly retractive.

Valdivia in [24, Page 161] showed that an (LF)-space $E=\mathop {\mathrm{ind\,}}_n E_n$ satisfies the condition ($M_0$) if, and only if, for all $m\in {{\mathbb {N}}}$ there is an absolutely convex 0-neighborhood $U_m$ of $E_m$ with $U_m\subseteq U_{m+1}$ such that, given any $n\in {{\mathbb {N}}}$ there is an integer $\mu > n$ such that $\sigma (E,E')$ and $\sigma (E_\mu ,E'_\mu )$ coincide on $U_m$.

2.2 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 ${{\mathcal {B}}}(X)$ 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 $(B_j)_{j\in {{\mathbb {N}}}}$ of G, there exists a sequence $(\lambda _j)_{j\in {{\mathbb {N}}}}\in (0,\infty )^{{{\mathbb {N}}}}$ such that $\bigcup _{j=1}^\infty \lambda _jB_j\in {{\mathcal {B}}}(G)$.

• (2) Let G be a metrizable lcHs. Then for every family of precompact subets $(C_j)_{j\in {{\mathbb {N}}}}$ of G, there exists a sequence $(\lambda _j)_{j\in {{\mathbb {N}}}}\in (0,\infty )^{{{\mathbb {N}}}}$ such that $\bigcup _{j=1}^\infty \lambda _jC_j$ is precompact in G.

We use the previous lemma to give the following characterizations.

Proposition 2.3

Let $E=\mathop {\mathrm{ind\,}}_n E_n$ and $F=\mathop {\mathrm{ind\,}}_n F_n$ be two (LF)-spaces. The following assertions hold:

• (1) Assume that F is regular. Then the linear operator $T:E\rightarrow F$ is bounded if, and only if, there exists $n\in {{\mathbb {N}}}$ such that for all $m\in {{\mathbb {N}}}$ we have that $T(E_m)\subset F_n$ and the restriction $T:E_m\rightarrow F_n$ is bounded.

• (2) Assume that F satisfies the condition (M). Then the linear operator $T:E\rightarrow F$ is compact if, and only if, there exists $n\in {{\mathbb {N}}}$ such that for all $m\in {{\mathbb {N}}}$ we have that the restriction $T:E_m\rightarrow F_n$ is compact.

.

Proof

Both the proofs are analogous. Hence, we show the claim for the compactness.

Suppose that $T:E\rightarrow F$ is compact. By assumption, we can find a 0-neighborhood U of E such that T(U) is relatively compact in F. Note that in F, since it is complete, precompactness and relative compactness are the same. Since F is equivalently precompactly retractive, we can find $n\in {{\mathbb {N}}}$ such that T(U) is precompact in $F_n$. For all $m\in {{\mathbb {N}}}$ the set $U':=U\cap E_m$ is a 0-neighborhood in $E_m$ such that $T(U')\subset T(U)\subset F_n$. Since T(U) is precompact in $F_n$, the same also holds for $T(U')$. This means that the map $T:E_m\rightarrow F_n$ is compact.

We assume now that the condition is fulfilled and prove that $T:E\rightarrow F$ is compact. By assumption, there exists $n\in {{\mathbb {N}}}$ such that for all $m\in {{\mathbb {N}}}$ we can find a 0-neighborhood $U_m$ in $E_m$ such that $T(U_m)$ is precompact in $F_n$. Now we apply Lemma 2.2 at the Fréchet space $F_n$ to find a sequence $(\lambda _m)_{m\in {{\mathbb {N}}}}\in (0,\infty )^{{{\mathbb {N}}}}$ such that $\bigcup _{m=1}^\infty \lambda _mT(U_m)$ is precompact in $F_n$. Set $U:=\Gamma \left( \bigcup _{m=1}^\infty \lambda _mU_m\right)$, where $\Gamma$ denotes the absolutely convex hull of the union. Clearly U is a 0-neighborhood of E satisfying $T(U)\subset \Gamma \left( \bigcup _{m=1}^\infty \lambda _mT(U_m)\right)$, 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. $\square$

Given X, Y two lcHs, a linear operator $T : X \rightarrow Y$ is called Montel if it maps bounded subsets of X into relatively compact subsets of Y. If X and Y are Banach spaces, then $T:X\rightarrow Y$ is Montel if, and only if, it is compact. For an operator between (LF)-spaces we have the following result.

Proposition 2.4

Let $E=\mathop {\mathrm{ind\,}}_n E_n$ and $F=\mathop {\mathrm{ind\,}}_n F_n$ be two (LF)-spaces. Suppose that F satisfies the condition (M), and E is regular. Then the continuous linear operator $T:E\rightarrow F$ is Montel if, and only if, for all $m\in {{\mathbb {N}}}$ there exists $n\in {{\mathbb {N}}}$ such that the restriction $T:E_m\rightarrow F_n$ is Montel.

Proof

Suppose that $T:E\rightarrow F$ is Montel. Fixed $m\in {{\mathbb {N}}}$, by the continuity of T there exists $n\in {{\mathbb {N}}}$ such that $T:E_m\rightarrow F_n$ is continuous (see [14]). Since F is in particular strongly boundedly retractive, we choose a $n'\ge n$ as in the definition of the condition and prove that $T:E_m\rightarrow F_{n'}$ 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 $T:E\rightarrow F$ is Montel. Fix a bounded subset B of E. Due to the regularity of E, we can find $m\in {{\mathbb {N}}}$ such that B is bounded in $E_m$. By assumption, there exists $n\in {{\mathbb {N}}}$ such that the restriction $T:E_m\rightarrow F_n$ is Montel. Hence, T(B) is relatively compact in $F_n$ and so in F. $\square$

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 $T : X \rightarrow Y$ 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 $T:X \rightarrow Y$ is reflexive. We refer the reader to [16] for more details.

We give the following characterization concerning (LF)-spaces.

Proposition 2.6

Let $E=\mathop {\mathrm{ind\,}}_n E_n$ and $F=\mathop {\mathrm{ind\,}}_n F_n$ be two (LF)-spaces. Suppose that F satisfies the condition ($M_0$), and E is regular. Then the continuous linear operator $T:E\rightarrow F$ is reflexive if, and only if, for all $m\in {{\mathbb {N}}}$ there exists $n\in {{\mathbb {N}}}$ such that the restriction $T:E_m\rightarrow F_n$ is reflexive.

Proof

Suppose that $T:E\rightarrow F$ is reflexive. Fixed $m\in {{\mathbb {N}}}$, by the continuity of T there exists $n\in {{\mathbb {N}}}$ such that $T:E_m\rightarrow F_n$ 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 $(U_k)_{k\in {{\mathbb {N}}}}$ of subsets of F such that $U_k$ is an absolutely convex 0-neighborhood of $F_k$ for all $k\in {{\mathbb {N}}}$ and the topologies induced on $U_k$ from $\sigma (F,F')$ and $\sigma (F_{n'},F'_{n'})$ coincide, for some $n'> k$. From the boundedness of T(B), we can find $\lambda >0$ such that $T(B)\subset \lambda U_n$. 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 $T:E_m\rightarrow F_{n'}$ is reflexive.

We assume now that the condition is fulfilled and prove that $T:E\rightarrow F$ is reflexive. Fix a bounded subset B of E. Due to the regularity of E, we can find $m\in {{\mathbb {N}}}$ such that B is bounded in $E_m$. By assumption, there exists $n\in {{\mathbb {N}}}$ such that $T:E_m\rightarrow F_n$ is reflexive. Hence, T(B) is relatively weakly compact in $F_n$ and so in F. $\square$

3 The sequence (LF)-spaces $l_p({\mathcal {V}})$

In this section, we introduce the sequence (LF)-spaces $l_p({\mathcal {V}})$ and recall the main properties of these spaces concerning regularity and completeness.

For all $n\in {{\mathbb {N}}}$, $V_n=\left( v_{n,k}\right) _{k\in {{\mathbb {N}}}}$ is a countable family of (strictly) positive sequences, called weights, on ${{\mathbb {N}}}$. We denote by ${\mathcal {V}}$ the sequence $\left( V_{n}\right) _{n\in {{\mathbb {N}}}}$ and we assume that the following two conditions are satisfied:

• (1) $v_{n,k}(i)\le v_{n,k+1}(i)$ for all $n,k\in {{\mathbb {N}}}$ and $i\in {{\mathbb {N}}}$;

• (2) $v_{n,k}(i)\ge v_{n+1,k}(i)$ for all $n,k\in {{\mathbb {N}}}$ and $i\in {{\mathbb {N}}}$.

Given a system of weights ${\mathcal {V}}$ as above, for $n,k\in {{\mathbb {N}}}$ and $1\le p\le \infty$ we define as usual

$$\begin{aligned} l_p(v_{n,k}):=\left\{ x=(x_i)_{i\in {{\mathbb {N}}}}\in \omega \, |\, p_{v_{n,k}}(x):=\Vert (x_iv_{n,k}(i))_{i\in {{\mathbb {N}}}}\Vert _{p}<\infty \right\} , \end{aligned}$$

where $\Vert \cdot \Vert _{p}$ denotes the usual $l_p$ norm. For $p=0$ we set

$$\begin{aligned} c_0(v_{n,k}):=\left\{ x=(x_i)_{i\in {{\mathbb {N}}}}\in \omega \, |\, \lim _{i\rightarrow \infty } v_{n,k}(i) x_i=0 \right\} . \end{aligned}$$

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 $l_\infty (v_{n,k})$. Since $l_p(v_{n,k+1})$ is continuously embedded into $l_p(v_{n,k})$, the sequence $\{l_p(v_{n,k})\}_{k\in {{\mathbb {N}}}}$ of Banach spaces forms a projective spectrum. Hence, for all $n\in {{\mathbb {N}}}$ and $1\le p\le \infty$, we can consider the echelon spaces

$$\begin{aligned} \lambda _p(V_n):=\bigcap _{k\in {{\mathbb {N}}}} l_p(v_{n,k})\; \mathrm{and} \; \lambda _0(V_n):=\bigcap _{k\in {{\mathbb {N}}}} c_0(v_{n,k}). \end{aligned}$$

Endowed with the projective topologies $\lambda _p(V_n)=\mathop {\mathrm{proj\,}}_{k} l_p(v_{n,k})$ (resp. $\lambda _0(V_n)= \mathop {\mathrm{proj\,}}_{k} c_0(v_{n,k})$), these spaces are Fréchet with the topology defined by the corresponding seminorms $p_{n,k}:=p_{v_{n,k}}$, $k\in {{\mathbb {N}}}$.

Condition (2) implies that $\lambda _p(V_n)$ is continuously embedded into $\lambda _p(V_{n+1})$. Hence, the sequence $\{\lambda _p(V_n)\}_{n\in {{\mathbb {N}}}}$ for $1\le p\le \infty \cup \{0\}$ of Fréchet spaces forms an inductive spectrum and the spaces

$$\begin{aligned} l_p({\mathcal {V}}):=\bigcup _{n\in {{\mathbb {N}}}} \lambda _p(V_n)\; \mathrm{and} \;l_0({\mathcal {V}}):=\bigcup _{n\in {{\mathbb {N}}}} \lambda _0(V_n) \end{aligned}$$

endowed with the inductive topologies $l_p({\mathcal {V}}):=\mathop {\mathrm{ind\,}}_{n} \lambda _p(V_n)$ (resp. $l_0({\mathcal {V}}):=\mathop {\mathrm{ind\,}}_{n} \lambda _0(V_n)$) are (LF)-spaces.

To describe the inductive topology of these spaces, we can associate to $l_p({\mathcal {V}})$ the Nachbin family in the usual way (see [8, 24]). Set

$$\begin{aligned} {\overline{V}}:=\left\{ {\overline{v}}=({\overline{v}}_i)_{i\in {{\mathbb {N}}}}\in (0,\infty )^{{\mathbb {N}}}\,|\,\forall n\in {{\mathbb {N}}}\, \exists k=k(n)\in {{\mathbb {N}}}: \sup _{i\in {{\mathbb {N}}}}\frac{{\overline{v}}_i}{v_{n,k}(i)}<\infty \right\} . \end{aligned}$$

A sequence ${\overline{v}}$ belongs to ${\overline{V}}$ if, and only if, ${\overline{v}}_i=\inf _{n\in {{\mathbb {N}}}} \alpha _n v_{n, k(n)}(i)$, for some $\alpha _n>0$, $k(n)\in {{\mathbb {N}}}$ and for all $i\in {{\mathbb {N}}}$. For $1\le p\le \infty$ we define in the usual way the Banach spaces

$$\begin{aligned} l_p({\overline{v}}):=\{x=(x_i)_{i\in {{\mathbb {N}}}}\,|\, \Vert ({\overline{v}}_ix_i)_{i\in {{\mathbb {N}}}}\Vert _p<\infty \}\; \mathrm{and} \; c_0({\overline{v}}):=\left\{ x=(x_i)_{i\in {{\mathbb {N}}}}\, |\, \lim _{i\rightarrow \infty } {\overline{v}}_ix_i=0 \right\} . \end{aligned}$$

For $1\le p\le \infty$ or $p=0$ we denote by

$$\begin{aligned} K_p({\overline{V}}):=\mathop {\mathrm{proj\,}}_{{\mathop {{\overline{v}}\in {\overline{V}}}\limits ^{\leftarrow }}} l_p({\overline{v}})\; \mathrm{and} \; K_0({\overline{V}}):= \mathop {\mathrm{proj\,}}_{{\mathop {{\overline{v}}\in {\overline{V}}}\limits ^{\leftarrow }}} c_0({\overline{v}}). \end{aligned}$$

The spaces $K_p({\overline{V}})$, for $1\le p\le \infty \, \cup \{0\}$, are complete, being a projective limit of complete spaces, and their seminorms will be denoted by $p_{{\overline{v}}}$, ${\overline{v}}\in {\overline{V}}$, when p is fixed and no confusion shall arise.

Remark 3.1

The inclusion $l_p({\mathcal {V}})\hookrightarrow K_p({\overline{V}})$, for $1\le p<\infty \,\cup \{0\}$, is a topological isomorphism into as a consequence of [24, Proposition 5.1].

For $p=\infty$, the inclusion $l_\infty ({\mathcal {V}})\hookrightarrow K_\infty ({\overline{V}})$ also holds. We need to require, in addition, that the system of weights ${\overline{V}}$ satisfies the following condition (see [8, Theorem 7]): $\forall (k(n))_{n\in {{\mathbb {N}}}}\subset {{\mathbb {N}}}$ $\exists {\overline{v}}\in {\overline{V}}$ $\forall m\in {{\mathbb {N}}}, {\overline{w}}_m\in {\overline{W}}_m$ $\exists M\in {{\mathbb {N}}}$ such that

$$\begin{aligned} \min \left( {\overline{w}}_m,{\overline{v}}^{-1}\right) \le \sum _{n=1}^{M} v_{n,k(n)}^{-1}, \end{aligned}$$

(3.1)

where ${\overline{W}}_m$ denotes the system of all non-negative sequences which are dominated by sequences of the form $\inf _{k\in {{\mathbb {N}}}} \alpha _k v_{m,k(m)}^{-1}$, for some $\alpha _k>0$, $k(m)\in {{\mathbb {N}}}$. In the (LB)-case, condition (3.1) is equivalent to condition (D) of Bierstedt and Meise (see [7]).

Furthermore, the inclusion $l_p({\mathcal {V}})\hookrightarrow K_p({\overline{V}})$, for $1\le p<\infty \cup \,\{0\}$, is also with dense range. This means that $K_p({\overline{V}})=\widehat{l_p({\mathcal {V}})}$, where $\widehat{l_p({\mathcal {V}})}$ stands for the topological completion of $l_p({\mathcal {V}})$.

In the following, we recall the conditions for which we have that $l_p({\mathcal {V}})=K_p({\overline{V}})$ algebraically and also topologically. Due to Remark 3.1, $l_p({\mathcal {V}})=K_p({\overline{V}})$ if, and only if, the (LF)-space $l_p({\mathcal {V}})$ is complete.

In our context, the characterization of the regularity is due to Vogt [24], in terms of a condition on the system of weights ${\mathcal {V}}$.

Definition 3.2

We say that the sequence ${\mathcal {V}}=\left( \left( v_{n,k} \right) _{k\in {{\mathbb {N}}}} \right) _{n\in {{\mathbb {N}}}}$ satisfies the condition (WQ) (or is of type (WQ)) if

$$\begin{aligned} \forall n\in {{\mathbb {N}}}\, \exists \mu ,&m\in {{\mathbb {N}}}\, \forall k, N\in {{\mathbb {N}}}, \, \exists K\in {{\mathbb {N}}}, S>0,\, s.t. \forall i\in {{\mathbb {N}}}: \\&v_{m,k}(i)\le S(v_{n,\mu }(i)+ v_{N,K}(i)). \end{aligned}$$

Definition 3.3

We say that the sequence ${\mathcal {V}}=\left( \left( v_{n,k} \right) _{k\in {{\mathbb {N}}}} \right) _{n\in {{\mathbb {N}}}}$ satisfies the condition (Q) (or is of type (Q)) if

$$\begin{aligned} \forall n\in {{\mathbb {N}}}\, \exists \mu ,&m\in {{\mathbb {N}}}\, \forall k, N\in {{\mathbb {N}}}, R>0 \, \exists K\in {{\mathbb {N}}}, S>0,\, s.t. \forall i\in {{\mathbb {N}}}: \\&v_{m,k}(i)\le \frac{1}{R}v_{n,\mu }(i)+ Sv_{N,K}(i). \end{aligned}$$

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 $l_p({\mathcal {V}})$ in the following theorem of Vogt (see [24]).

Theorem 3.4

• (1) For $1<p<\infty$, the following conditions are equivalent:

  • (i) ${\mathcal {V}}$ satisfies the condition (WQ);

  • (ii) $l_p({\mathcal {V}})$ is regular;

  • (iii) $l_p({\mathcal {V}})$ is complete;

  • (iv) $l_p({\mathcal {V}})$ is reflexive.

• (2) For $p=1,\infty$, the following conditions are equivalent:

  • (i) ${\mathcal {V}}$ satisfies the condition (WQ);

  • (ii) $l_p({\mathcal {V}})$ is regular;

  • (iii) $l_p({\mathcal {V}})$ is complete.

• (3) For $p=0$, the following conditions are equivalent:

  • (i) ${\mathcal {V}}$ satisfies the condition (Q);

  • (ii) $l_0({\mathcal {V}})$ is regular;

  • (iii) $l_0({\mathcal {V}})$ is complete.

By Remark 3.1 and Theorem 3.4 (see also [8]), we have

$$\begin{aligned}&l_0({\mathcal {V}})=K_0({\overline{V}}) \, \textit{alg. and top.}\; \Leftrightarrow \; {\mathcal {V}} \, \textit{satisfies the condition}\;( Q) \end{aligned}$$

(3.2)

$$\begin{aligned}&l_\infty ({\mathcal {V}})=K_\infty ({\overline{V}}) \, \textit{alg. and top.}\; \Leftrightarrow \; {\mathcal {V}} \, \textit{satisfies the conditions}\;(WQ)+(3.1) \end{aligned}$$

(3.3)

$$\begin{aligned}&l_p({\mathcal {V}})=K_p({\overline{V}}) \, \textit{alg. and top.}\; 1\le p<\infty \;\Leftrightarrow \; {\mathcal {V}} \, \textit{satisfies the condition}\;(WQ) . \end{aligned}$$

(3.4)

This result means that the inductive limit topology of the (LF)-spaces $l_p({\mathcal {V}})$, under the above conditions (3.2), (3.3), (3.4), coincide with the lc’one induced by the family of seminorms $(p_{{\overline{v}}})_{{\overline{v}}\in {\overline{V}}}$.

4 Diagonal operators acting on $l_p({\mathcal {V}})$

From now on, we work with diagonal (multiplication) operators on the sequence (LF)-spaces $l_p({\mathcal {V}})$. We denote by $\omega$ the space of all the sequences $\varphi =(\varphi _i)_{i\in {{\mathbb {N}}}}\in {{\mathbb {C}}}^{{\mathbb {N}}}$. Given a sequence $\varphi =(\varphi _i)_{i\in {{\mathbb {N}}}}\in \omega$, we define the multiplication (or diagonal) operator as $M_\varphi :\omega \rightarrow \omega$ such that $(x_i)_{i\in {{\mathbb {N}}}}\mapsto (x_i\varphi _i)_{i\in {{\mathbb {N}}}}$. If $M_\varphi$ acts continuously from a sequence lcHs X into a sequence lcHs Y, we say that $\varphi$ is a multiplier from X to Y.

Firstly, we characterize the multipliers between the sequence (LF)-spaces $l_p({\mathcal {V}})$.

Theorem 4.1

Let ${\mathcal {V}}, {\mathcal {W}}$ be two systems of weights and fix $\varphi =(\varphi _i)_{i\in {{\mathbb {N}}}}\in \omega$. For $1\le p\le \infty \cup \{0\}$, the following properties are equivalent:

• (1) $M_\varphi :l_p({\mathcal {V}})\rightarrow l_p({\mathcal {W}})$ is well-defined;

• (2) $M_\varphi :l_p({\mathcal {V}})\rightarrow l_p({\mathcal {W}})$ is continuous;

• (3) For all $m\in {{\mathbb {N}}}$ there exists $n \in {{\mathbb {N}}}$ such that for all $k\in {{\mathbb {N}}}$ there exist $l\in {{\mathbb {N}}}$ for which

  $$\begin{aligned} \sup _{i\in {{\mathbb {N}}}} \frac{w_{n,k}(i)|\varphi _i|}{v_{m,l}(i)}<\infty . \end{aligned}$$

  (4.1)

Proof

Clearly, (2) implies (1) and (1) implies (2) by the Closed Graph Theorem [16, Page 57]. Indeed, if $x=(x_i)_{i}\subset l_p({\mathcal {V}})$ is a net convergent to ${\overline{x}}$ in $l_p({\mathcal {V}})$ and $(M_\varphi (x_i))_{i}$ is convergent to ${\overline{y}}$ in $l_p({\mathcal {W}})$, then ${\overline{y}}=\varphi {\overline{x}}=M_\varphi ({\overline{x}})$. This proves that the graph of $M_\varphi$ is closed.

We prove the statement for $1\le p\le \infty$. The proof in the case $p=0$ is analogous and so is omitted.

(2)$\Leftrightarrow$(3). The diagonal operator $M_\varphi :l_p({\mathcal {V}})\rightarrow l_p({\mathcal {W}})$ is continuous if, and only if, for all $m\in {{\mathbb {N}}}$ the diagonal operator $M_\varphi :\lambda _p(V_m)\rightarrow l_p({\mathcal {W}})$ is continuous. From Grothendieck’s Factorization Theorem [14, Page 147], $M_\varphi$ is continuous if, and only if, for all $m\in {{\mathbb {N}}}$ there exists $n\in {{\mathbb {N}}}$ such that $M_\varphi :\lambda _p(V_m)\rightarrow \lambda _p(W_n)$ is well-defined and continuous. Now the diagonal operator between the echelon spaces is continuous if, and only if, for all $k\in {{\mathbb {N}}}$ there exists $l\in {{\mathbb {N}}}$ such that $M_\varphi :l_p(v_{m,l})\rightarrow l_p(w_{n,k})$ is continuos. Consider the isometries $M_{v_{m,l}}:l_p(v_{m,l})\rightarrow l_p$ such that $(x_i)_{i\in {{\mathbb {N}}}}\mapsto (v_{m,l}(i)x_i)_{i\in {{\mathbb {N}}}}$ and $M_{w_{n,k}}:l_p(w_{n,k})\rightarrow l_p$ such that $(x_i)_{i\in {{\mathbb {N}}}}\mapsto (w_{n,k}(i)x_i)_{i\in {{\mathbb {N}}}}$. Setting $\phi := \left( \frac{w_{n,k}(i)\varphi _i}{v_{m,l}(i)}\right) _{i\in {{\mathbb {N}}}}$, then $M_\phi = M_{w_{n,k}}\circ M_\varphi \circ M_{v_{m,l}}^{-1}$. Taking into account that $M_{\phi } :l_p\rightarrow l_p$ is continuous if, and only if, $\phi$ is bounded (see [4, Lemma 15]), we get (4.1).

$\square$

A similar characterization holds for the boundedness of multiplication operators between the sequence (LF)-spaces $l_p({\mathcal {V}})$. We need to recall the characterization of boundedness for operators acting between Fréchet spaces (see [4, Lemma 25]).

Lemma 4.2

Let $E=\mathop {\mathrm{proj\,}}_m E_m$ and $F=\mathop {\mathrm{proj\,}}_m F_m$ be Fréchet spaces such that $E=\bigcap _{m=1}^\infty E_m$, with each $(E_m,\Vert \cdot \Vert _m)$ a Banach space (resp. $F=\bigcap _{m=1}^\infty F_m$, with each $(F_m,\Vert \cdot \Vert _m)$ a Banach space). Moreover, it is assumed that E is dense in $E_m$ and that $E_m \hookrightarrow E_{m+1}$ for all $m\in {{\mathbb {N}}}$ (resp. $F_m \hookrightarrow F_{m+1}$ for all $m\in {{\mathbb {N}}}$). Then the continuous linear operator $T:E\rightarrow F$ is bounded if, and only if, there exists $l\in {{\mathbb {N}}}$ such that for all $k\in {{\mathbb {N}}}$ the operator T has a unique continuous linear extension $T:E_l\rightarrow F_k$.

Remark 4.3

We observe that Lemma 4.2 continues to hold even if T is not assumed to be continuous. The result follows with the same proof contained in [4, Lemma 25].

Theorem 4.4

Let ${\mathcal {V}}, {\mathcal {W}}$ be two systems of weights and fix $\varphi =(\varphi _i)_{i\in {{\mathbb {N}}}}\in \omega$. For $1\le p<\infty \, \cup \{0\}$, assume that $l_p({\mathcal {W}})$ is regular. Then $M_\varphi$ is bounded if, and only if, there exists $n\in {{\mathbb {N}}}$ such that for all $m\in {{\mathbb {N}}}$ there exists $l \in {{\mathbb {N}}}$ such that for all $k\in {{\mathbb {N}}}$

$$\begin{aligned} \sup _{i\in {{\mathbb {N}}}} \frac{w_{n,k}(i)|\varphi _i|}{v_{m,l}(i)}<\infty . \end{aligned}$$

(4.2)

Proof

By Proposition 2.3 (1), $M_\varphi :l_p({\mathcal {V}})\rightarrow l_p({\mathcal {W}})$ is bounded if, and only if, there exists $n\in {{\mathbb {N}}}$ such that for all $m\in {{\mathbb {N}}}$ the restriction $M_\varphi :\lambda _p(V_m)\rightarrow \lambda _p(W_n)$ is bounded. Using Lemma 4.2, this holds if, and only if, there exists $l \in {{\mathbb {N}}}$ such that for all $k\in {{\mathbb {N}}}$ the operator $M_\varphi$ has a unique continuous linear extension $M_\varphi :l_p(v_{m,l})\rightarrow l_p(w_{n,k})$. As in the proof of Theorem 4.1, this is equivalent to requiring that (4.2) is satisfied.

The same also holds for $p=0$. $\square$

The next target is to describe when a diagonal operator between the sequence (LF)-spaces $l_p({\mathcal {V}})$ 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 $\phi \in \omega$. The following assertions hold:

• (1) The multiplication operator $M_\phi :c_0\rightarrow c_0$ is compact if, and only if, $\phi \in c_0$.

• (2) For $1\le p<\infty$, the multiplication operator $M_\phi :l_p\rightarrow l_p$ is compact if, and only if, $\phi \in c_0$.

As for the boundedness, we also need a characterization of the compactness for operators acting between Fréchet spaces. Again we refer to [4].

Lemma 4.6

Let $E=\mathop {\mathrm{proj\,}}_m E_m$ and $F=\mathop {\mathrm{proj\,}}_m F_m$ be Fréchet spaces such that $E=\bigcap _{m=1}^\infty E_m$, with each $(E_m,\Vert \cdot \Vert _m)$ a Banach space (resp. $F=\bigcap _{m=1}^\infty F_m$, with each $(F_m,\Vert \cdot \Vert _m)$ a Banach space). Moreover, it is assumed that E is dense in $E_m$ and that $E_m \hookrightarrow E_{m+1}$ for all $m\in {{\mathbb {N}}}$ (resp. $F_m \hookrightarrow F_{m+1}$ for all $m\in {{\mathbb {N}}}$). Then the linear operator $T:E\rightarrow F$ is compact if, and only if, there exists $l\in {{\mathbb {N}}}$ such that for all $k\in {{\mathbb {N}}}$ the operator T has a unique compact linear extension $T:E_l\rightarrow F_k$.

Theorem 4.7

Let ${\mathcal {V}}, {\mathcal {W}}$ be two systems of weights and fix $\varphi =(\varphi _i)_{i\in {{\mathbb {N}}}}\in \omega$. For $1\le p<\infty \, \cup \{0\}$, assume that $l_p({\mathcal {W}})$ satisfies the condition (M). Then $M_\varphi$ is compact if, and only if, there exists $n\in {{\mathbb {N}}}$ such that for all $m\in {{\mathbb {N}}}$ there exists $l \in {{\mathbb {N}}}$ such that for all $k\in {{\mathbb {N}}}$

$$\begin{aligned} \lim _{i\rightarrow \infty } \frac{w_{n,k}(i)|\varphi _i|}{v_{m,l}(i)}=0. \end{aligned}$$

(4.3)

Proof

By Proposition 2.3 (2), $M_\varphi :l_p({\mathcal {V}})\rightarrow l_p({\mathcal {W}})$ is compact if, and only if, there exists $n\in {{\mathbb {N}}}$ such that for all $m\in {{\mathbb {N}}}$ the restriction $M_\varphi :\lambda _p(V_m)\rightarrow \lambda _p(W_n)$ is compact. Using Lemma 4.6, this holds if, and only if, there exists $l \in {{\mathbb {N}}}$ such that for all $k\in {{\mathbb {N}}}$ the extension $M_\varphi :l_p(v_{m,l})\rightarrow l_p(w_{n,k})$ is compact. We prove that it is equivalent to the limit (4.3). Consider the isometries $M_{v_{m,l}}:l_p(v_{m,l})\rightarrow l_p$ such that $(x_i)_{i\in {{\mathbb {N}}}}\mapsto (v_{m,l}(i)x_i)_{i\in {{\mathbb {N}}}}$ and $M_{w_{n,k}}:l_p(w_{n,k})\rightarrow l_p$ such that $(x_i)_{i\in {{\mathbb {N}}}}\mapsto (w_{n,k}(i)x_i)_{i\in {{\mathbb {N}}}}$. Setting $\phi := \left( \frac{w_{n,k}(i)\varphi _i}{v_{m,l}(i)}\right) _{i\in {{\mathbb {N}}}}$, then $M_\phi = M_{w_{n,k}}\circ M_\varphi \circ M_{v_{m,l}}^{-1}$. Thus, $M_\varphi$ is compact if, and only if, $M_\phi :l_p\rightarrow l_p$ is compact. Due to Lemma 4.5, this holds if, and only if, $\phi \in c_0$, i.e., if $\phi$ vanishes at infinity.

The same also holds for $p=0$. $\square$

To describe when diagonal operators between the sequence (LF)-spaces $l_p({\mathcal {V}})$ 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 $1\le p<\infty$, a subset K of $l_p$ is relatively compact if, and only if, for every $\varepsilon >0$ there exists $j_0\in {{\mathbb {N}}}$ such that for every $x\in K$ we have $\sum _{j=j_0+1}^{\infty }|x_j|^p<\varepsilon ^p$. Analogously, for $p=0$ a subset K of $c_0$ is relatively compact if, and only if, for every $\varepsilon >0$ there exists $j_0\in {{\mathbb {N}}}$ such that for every $x\in K$ we have $\sup _{j\ge j_0+1}|x_j|<\varepsilon$.

Let $A=(a_n)_{n\in {{\mathbb {N}}}}$ denote a Köthe matrix, i.e., an increasing sequence of strictly positive functions $a_n$. We consider

$$\begin{aligned} \lambda _\infty (A)_+=\{x=(x_i)_{i\in {{\mathbb {N}}}}\in \omega \,|\,\Vert (a_n x)_{n\in {{\mathbb {N}}}}\Vert _\infty <\infty \,\text {and}\, x_i>0\, \forall i\in {{\mathbb {N}}}\}. \end{aligned}$$

The following useful description of the bounded sets in a Köthe echelon space is due to Bierstedt et al. [9].

Proposition 4.9

Let $A=(a_n)_{n\in {{\mathbb {N}}}}$ be a Köthe matrix. For $1\le p<\infty \, \cup \{0\}$, a subset B of $\lambda _p(A)$ is bounded if, and only if, there exists ${\overline{a}}\in \ \lambda _\infty (A)_+$ such that $B\subseteq B_{{\overline{a}}}:=\left\{ x\in \omega \,|\, \left\| \left( \frac{x_i}{{\overline{a}}(i)}\right) _{i\in {{\mathbb {N}}}}\right\| _p \le 1 \right\}$.

First of all, we study the case $p<\infty$. Fixed a weight v, we recall that $c_0$ is isomorphic to $c_0(v)$ through to the map $(x_i)_{i\in {{\mathbb {N}}}}\mapsto \left( \frac{x_i}{v(i)}\right) _{i\in {{\mathbb {N}}}}$. Hence, taking into account Remark 4.8, a subset K of $c_0(v)$ is relatively compact if, and only if, for every $\varepsilon >0$ there exists $j_0\in {{\mathbb {N}}}$ such that for every $x\in K$ we have $\sup _{j\ge j_0+1}v(j)|x_j|<\varepsilon$. Analogously for $1\le p <\infty$.

Proposition 4.10

Let $A=(a_n)_{n\in {{\mathbb {N}}}}$, $B=(b_m)_{m\in {{\mathbb {N}}}}$ be two Köthe matrices and fix $\varphi =(\varphi _i)_{i\in {{\mathbb {N}}}}\in \omega$. For $1\le p<\infty \, \cup \{0\}$, assume that $M_\varphi :\lambda _p(A)\rightarrow \lambda _p(B)$ is continuous. The diagonal operator $M_\varphi$ is Montel if, and only if, for every ${\overline{a}}\in \lambda _\infty (A)_+$ and for all $m\in {{\mathbb {N}}}$

$$\begin{aligned} \lim _{i\rightarrow \infty } {\overline{a}}(i)|\varphi _i|b_m(i)=0. \end{aligned}$$

(4.4)

Proof

Observe that from [22, Lemma 1], $M_\varphi :\lambda _p(A)\rightarrow \lambda _p(B)$ is continuous for $1\le p<\infty$ if, and only if, $M_\varphi :\lambda _0(A)\rightarrow \lambda _0(B)$ is continuous. We prove the statement for $p=0$. The proof in the case $1\le p<\infty$ is analogous and so is omitted.

Suppose that $M_\varphi :\lambda _0(A)\rightarrow \lambda _0(B)$ is Montel. Given ${\overline{a}}\in \lambda _\infty (A)_+$ and $m\in {{\mathbb {N}}}$, the set $B_{{\overline{a}}}$ is bounded in $\lambda _0(A)$ due to Proposition 4.9. By assumption, $M_\varphi$ is Montel, so $M_\varphi (B_{{\overline{a}}})$ is relatively compact in $\lambda _0(B)$ and hence in $c_0(b_m)$. Now it is easy to see that ${\overline{a}}(j)e_j\in B_{{\overline{a}}}$ for all $j\in {{\mathbb {N}}}$. Moreover, $M_\varphi (({\overline{a}}(j)e_j)_{j\in {{\mathbb {N}}}})=(\varphi _j{\overline{a}}(j)e_j)_{j\in {{\mathbb {N}}}}$. Furthermore, the sequence $(\varphi _j{\overline{a}}(j)e_j)_{j\in {{\mathbb {N}}}}$ converges to 0 coordinatewise in ${{\mathbb {C}}}^{{\mathbb {N}}}$. Since $M_\varphi (B_{{\overline{a}}})$ is relatively compact in $c_0(b_m)$, the topology of $c_0(b_m)$ on $M_\varphi (B_{{\overline{a}}})$ coincides with the topology induced by ${{\mathbb {C}}}^{{\mathbb {N}}}$. This means that the sequence $(\varphi _j{\overline{a}}(j)e_j)_{j\in {{\mathbb {N}}}}$ converges to 0 in $c_0(b_m)$ and so (4.4) holds ($\Vert e_j\Vert _{c_0(b_m)}=b_m(j)$).

We assume now that the condition is fulfilled and prove that $M_\varphi :\lambda _0(A)\rightarrow \lambda _0(B)$ is Montel. We want to prove that for a fixed bounded subset ${{\mathcal {B}}}$ of $\lambda _0(A)$ the set $M_\varphi ({{\mathcal {B}}})$ is relatively compact in $c_0(b_m)$ for all $m\in {{\mathbb {N}}}$. To do this, since ${{\mathcal {B}}}$ is bounded, we apply Proposition 4.9 and choose ${\overline{a}}\in \lambda _\infty (A)_+$ such that ${{\mathcal {B}}}\subset B_{{\overline{a}}}$. It suffices to show that $M_\varphi (B_{{\overline{a}}})$ is relatively compact in $c_0(b_m)$ for all $m\in {{\mathbb {N}}}$, that is, using Remark 4.8, for every $\varepsilon >0$ and $m\in {{\mathbb {N}}}$ there exists $j_0\in {{\mathbb {N}}}$ such that for every $y\in M_\varphi (B_{{\overline{a}}})$ we have $\sup _{j\ge j_0+1}b_m(j)|y_j|<\varepsilon$. Hence, fixed $m\in {{\mathbb {N}}}$ and $\varepsilon >0$, since (4.4) holds, we can choose $j_0\in {{\mathbb {N}}}$ such that ${\overline{a}}(j)|\varphi _j|b_m(j)<\varepsilon$ for all $j\ge j_0+1$. If $y\in M_\varphi (B_{{\overline{a}}})$, then $y=(y_i)_{i\in {{\mathbb {N}}}}=(\varphi _i x_i)_{i\in {{\mathbb {N}}}}$ and $|x_i|\le {\overline{a}}(i)$ for all $i\in {{\mathbb {N}}}$. Then, if $j\ge j_0+1$ and $y\in M_\varphi (B_{{\overline{a}}})$, we get

$$\begin{aligned} |y_j|b_m(j)\le {\overline{a}}(j)|\varphi _j|b_m(j)<\varepsilon . \end{aligned}$$

$\square$

Therefore, the case $p<\infty$ is characterized. Now we prove that the same characterization holds for $p=\infty$.

Proposition 4.11

Let $A=(a_n)_{n\in {{\mathbb {N}}}}$, $B=(b_m)_{m\in {{\mathbb {N}}}}$ be two Köthe matrices and fix $\varphi =(\varphi _i)_{i\in {{\mathbb {N}}}}\in \omega$. Assume that $M_\varphi :\lambda _\infty (A)\rightarrow \lambda _\infty (B)$ is continuous. Then $M_\varphi$ is Montel if, and only if, $M_\varphi :\lambda _0(A)\rightarrow \lambda _0(B)$ is Montel.

Proof

First of all, let observe that $M_\varphi :\lambda _\infty (A)\rightarrow \lambda _\infty (B)$ is continuous if, and only if, $M_\varphi :\lambda _0(A)\rightarrow \lambda _0(B)$ is continuous (see [22, Lemma 1]). Moreover, if $M_\varphi :\lambda _\infty (A)\rightarrow \lambda _\infty (B)$ is Montel and ${{\mathcal {B}}}$ is a bounded subset of $\lambda _0(A)$, then $M_\varphi ({{\mathcal {B}}})$ is relatively compact in $\lambda _\infty (B)$ and hence in $\lambda _0(B)$, since $\lambda _0(B)$ is a closed subspace of $\lambda _\infty (B)$ and $M_\varphi ({{\mathcal {B}}})\subset \lambda _0(B)$.

We suppose that $M_\varphi :\lambda _0(A)\rightarrow \lambda _0(B)$ is Montel. Applying [13, Corollary 2.3], then $M^t_\varphi :\lambda _0(A)_b'\rightarrow \lambda _0(B)_b'$ is Montel. Note that $\lambda _0(A)_b'$ and $\lambda _0(B)_b'$ are complete (LB)-spaces (see [6, Proposition 10]). Then applying [13, Corollary 2.4], we get that $M^{tt}_\varphi :(\lambda _0(A)_{b}')_b'\rightarrow (\lambda _0(B)_{b}')_b'$ is Montel. Since $M^{tt}_\varphi =M_\varphi$ on $\lambda _\infty (A)$ and $\lambda _\infty (A)$ is the bidual of $\lambda _0(A)$ (resp. $\lambda _\infty (B)$ is the bidual of $\lambda _0(B)$), we get the claim. $\square$

Thus, we can characterize when a diagonal operator between the sequence (LF)-spaces $l_p({\mathcal {V}})$ is Montel. The proof is an application of Propositions 2.4, 4.10 and 4.11.

Theorem 4.12

Let ${\mathcal {V}}$, ${\mathcal {W}}$ be two systems of weights and fix $\varphi =(\varphi _i)_{i\in {{\mathbb {N}}}}\in \omega$. For $1\le p\le \infty \,\cup \{0\}$ assume that $l_p({\mathcal {V}})$ is regular and $l_p({\mathcal {W}})$ satisfies the condition (M). Suppose that the diagonal operator $M_\varphi :l_p({\mathcal {V}})\rightarrow l_p({\mathcal {W}})$ is continuous. Then $M_\varphi$ is Montel if, and only if, for all $m\in {{\mathbb {N}}}$ there exists $n\in {{\mathbb {N}}}$ such that for every ${\overline{v}}_m\in \lambda _\infty (V_m)_+$ and for all $k\in {{\mathbb {N}}}$

$$\begin{aligned} \lim _{i\rightarrow \infty } {\overline{v}}_m(i)|\varphi _i|w_{n,k}(i)=0. \end{aligned}$$

Now we treat the reflexivity of diagonal operators.

To study the Köthe echelon case, we have to make a distinction. For $p=1,0,\infty$, we want to show that a diagonal operator acting between the echelon spaces is Montel if, and only if, it is reflexive. Let $A=(a_n)_{n\in {{\mathbb {N}}}}$, $B=(b_m)_{m\in {{\mathbb {N}}}}$ be two Köthe matrices and fix $\varphi =(\varphi _i)_{i\in {{\mathbb {N}}}}\in \omega$. For $p=1,0,\infty$, assume that $M_\varphi :\lambda _p(A)\rightarrow \lambda _p(B)$ is continuous (see [22, Lemma 1]). We have:

• (i) If $p=1$, by Schur’s Theorem (see [26, Page 136]) a bounded subset B of $\lambda _1(A)$ is weakly (relatively) compact if, and only if, it is (relatively) compact. So, $M_\varphi :\lambda _1(A)\rightarrow \lambda _1(B)$ is reflexive if, and only if, it is Montel;

• (ii) If $p=0$, we prove that $M_\varphi :\lambda _0(A)\rightarrow \lambda _0(B)$ is reflexive if, and only if, it is Montel. We only have to show that the condition is necessary. So, suppose that $M_\varphi :\lambda _0(A)\rightarrow \lambda _0(B)$ is reflexive. By a result of Grothendieck [14] (see also [16, Page 204]), this implies that $M_\varphi =M_\varphi ^{tt}$ maps $\lambda _\infty (A)=(\lambda _0(A)_{b}')_b'$ in $\lambda _0(B)$. In particular, we have that $M_\varphi (\lambda _\infty (A)_+)\subset \lambda _0(B)$. This is equivalent to requiring that for every ${\overline{a}}\in \lambda _\infty (A)_+$ and $m\in {{\mathbb {N}}}$

  $$\begin{aligned} \lim _{i\rightarrow \infty } {\overline{a}}(i)|\varphi _i|b_m(i)=0. \end{aligned}$$

  By Proposition 4.10, since (4.4) holds, we get the claim.

• (iii) If $p=\infty$, we have that $M_\varphi :\lambda _\infty (A)\rightarrow \lambda _\infty (B)$ is reflexive if, and only if, it is Montel. Indeed applying [13, Corollaries 2.3, 2.4], $M_\varphi :\lambda _\infty (A)\rightarrow \lambda _\infty (B)$ reflexive implies $M_\varphi :\lambda _0(A)\rightarrow \lambda _0(B)$ reflexive. From the above case (ii), we get that this is equivalent to being Montel.

Thus, we can give a first characterization. The proof is an application of Propositions 2.4, 2.6 and the above considerations.

Theorem 4.13

Let ${\mathcal {V}}$, ${\mathcal {W}}$ be two systems of weights and fix $\varphi =(\varphi _i)_{i\in {{\mathbb {N}}}}\in \omega$. For $p=1,0,\infty$, assume that $l_p({\mathcal {V}})$ is regular and $l_p({\mathcal {W}})$ satisfies the condition $(M_0)$. Suppose that the diagonal operator $M_\varphi :l_p({\mathcal {V}})\rightarrow l_p({\mathcal {W}})$ is continuous. Then $M_\varphi$ is reflexive if, and only if, $M_\varphi$ is Montel.

Proof

We only have to prove that if $M_\varphi$ is reflexive, then $M_\varphi$ is Montel. By Proposition 2.6, for all $m\in {{\mathbb {N}}}$ there exists $n\in {{\mathbb {N}}}$ such that the restriction $M_\varphi :\lambda _p(V_m)\rightarrow \lambda _p(W_n)$ is reflexive. But $M_\varphi :\lambda _p(V_m)\rightarrow \lambda _p(W_n)$ is reflexive if, and only if, it is Montel. Therefore, applying Proposition 2.4, the diagonal operator $M_\varphi$ is Montel (take into account Remark 2.5). $\square$

Now we consider the case $1<p<\infty$. Observe that since $\lambda _p(A)$ and $\lambda _p(B)$ are reflexive spaces [6, Proposition 9], trivially the diagonal operator $M_\varphi :\lambda _p(A)\rightarrow \lambda _p(B)$ is reflexive. 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 ${\mathcal {V}}$, ${\mathcal {W}}$ be two systems of weights and fix $\varphi =(\varphi _i)_{i\in {{\mathbb {N}}}}\in \omega$. For $1<p<\infty$, assume that $l_p({\mathcal {W}})$ is regular. Then the diagonal operator $M_\varphi :l_p({\mathcal {V}})\rightarrow l_p({\mathcal {W}})$ is continuous if, and only if, it is reflexive.

5 Spectrum of diagonal operators acting on $l_p({\mathcal {V}})$

Let X be a lcHs and let ${{\mathcal {L}}}(X)$ denote the space of all continuous linear operators from X into itself. Given $T\in {{\mathcal {L}}}(X)$, the resolvent set of T is defined by

$$\begin{aligned} \rho (T,X):=\{\lambda \in {{\mathbb {C}}}\, |\, \lambda I-T:X\rightarrow X \ \mathrm{is\ bijective\ and \ } (\lambda I-T)^{-1}\in {{\mathcal {L}}}(X)\} \end{aligned}$$

and the spectrum of T is defined by $\sigma (T,X):={{\mathbb {C}}}\setminus \rho (T,X)$. The point spectrum is defined by

$$\begin{aligned} \sigma _p(T, X):=\{\lambda \in {{\mathbb {C}}}\, |\, \lambda I-T \mathrm{\ is\ not \ injective}\}. \end{aligned}$$

Unlike for Banach spaces, it may happens that $\rho (T, X)=\emptyset$ or that $\rho (T, X)$ is not open in ${{\mathbb {C}}}$ (see, e.g., [3]). This is the reason why many authors consider the subset $\rho ^*(T, X)$ of $\rho (T, X)$ consisting of all $\lambda \in {{\mathbb {C}}}$ for which there exists $\delta >0$ such that $B_\delta (\lambda ):=\{\mu \in {{\mathbb {C}}}\,|\, |\mu -\lambda |<\delta \}\subseteq \rho (T, X)$ and the set $\{(\mu I-T)^{-1}\,|\, \mu \in B_\delta (\lambda )\}$ is equicontinuous in ${{\mathcal {L}}}(X)$. If X is a Fréchet space, then it suffices that this set is bounded in ${{\mathcal {L}}}_s (X)$, where ${{\mathcal {L}}}_s(X)$ denotes ${{\mathcal {L}}}(X)$ endowed with the strong operator topology. The advantage of $\rho ^*(T,X)$, whenever it is not empty, is that it is open and the resolvent map is holomorphic from $\rho ^*(T,X)$ into ${{\mathcal {L}}}_b(X)$ (see, e.g., [2, Proposition 3.4]), where ${{\mathcal {L}}}_b(X)$ denotes ${{\mathcal {L}}}(X)$ endowed with the topology of the uniform convergence on bounded subsets of X). Define the Waelbrock spectrum $\sigma ^*(T,X) := {{\mathbb {C}}}\setminus \rho ^*(T,X)$, which is a closed set containing $\sigma (T,X)$.

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 ${\mathcal {V}}$ be a system of weights and fix $\varphi =(\varphi _i)_{i\in {{\mathbb {N}}}}\in \omega$. For $1\le p\le \infty \,\cup \{0\}$ we have that

$$\begin{aligned} \sigma _p(M_\varphi , l_p({\mathcal {V}}))=\{\varphi _i\,|\, i\in {{\mathbb {N}}}\}. \end{aligned}$$

We determine the resolvent set of the diagonal operators acting between the sequence (LF)-spaces $l_p({\mathcal {V}})$. Compare with [22, Proposition 1] for Köthe echelon spaces.

Proposition 5.2

Let ${\mathcal {V}}$ be a system of weights and fix $\varphi =(\varphi _i)_{i\in {{\mathbb {N}}}}\in \omega$. For $1\le p\le \infty \,\cup \{0\}$, the following assertions are equivalent:

• (1) $\mu \in \rho (M_\varphi ,l_p({\mathcal {V}}))$;

• (2) For all $m\in {{\mathbb {N}}}$ there exists $n\in {{\mathbb {N}}}$ such that for all $k\in {{\mathbb {N}}}$ there exist $l\in {{\mathbb {N}}}$ for which

  $$\begin{aligned} \sup _{i\in {{\mathbb {N}}}} \frac{v_{n,k}(i)}{v_{m,l}(i)|\varphi _i-\mu |}<\infty . \end{aligned}$$

  (5.1)

Proof

Define $\phi :=(\phi _i)_{i\in {{\mathbb {N}}}}\in \omega$ such that $\phi _i:=\frac{1}{\varphi _i-\mu }$. Clearly, $M_\phi$ is the continuous inverse of $M_\varphi -\mu I$, whenever it exists. Applying Theorem 4.1, we obtain the equivalence of (1) and (2). $\square$

Now we determine the Waelbrock spectrum. In contrast to what happens for the diagonal operators acting between Köthe echelon spaces [22, Theorem 1], for the sequence (LF)-spaces $l_p({\mathcal {V}})$ we have to require the completeness of the inductive limit.

Theorem 5.3

Let ${\mathcal {V}}$ be a system of weights and fix $\varphi =(\varphi _i)_{i\in {{\mathbb {N}}}}\in \omega$, For $1\le p\le \infty \, \cup \{0\}$, suppose that $l_p({\mathcal {V}})=K_p({\overline{V}})$ algebraically and topologically (see (3.2), (3.3), (3.4)). Then

$$\begin{aligned} \sigma ^*(M_\varphi , l_p({\mathcal {V}}))=\overline{\sigma (M_\varphi , l_p({\mathcal {V}}))}=\overline{\{\varphi _i\,|\, i\in {{\mathbb {N}}}\}}. \end{aligned}$$

(5.2)

Proof

Firstly, we prove the second equality of (5.2). By Lemma 5.1, we have that

$$\begin{aligned} \{\varphi _i\,|\, i\in {{\mathbb {N}}}\}=\sigma _p(M_\varphi , l_p({\mathcal {V}}))\subseteq \sigma (M_\varphi , l_p({\mathcal {V}}))\subseteq \overline{\sigma (M_\varphi , l_p({\mathcal {V}}))}, \end{aligned}$$

and hence $\overline{\{\varphi _i\,|\, i\in {{\mathbb {N}}}\}}\subseteq \overline{\sigma (M_\varphi , l_p({\mathcal {V}}))}$. For the other inclusion, let $\mu \notin \overline{\{\varphi _i\,|\, i\in {{\mathbb {N}}}\}}$. Then there is $\delta >0$ such that $|\varphi _i-\mu |>2\delta$ for all $i\in {{\mathbb {N}}}$. Therefore, arguing as in the proof of Proposition 5.2, we deduce that $\mu \in \rho (M_\varphi ,l_p({\mathcal {V}}))$. Suppose that $\mu \in \overline{\sigma (M_\varphi , l_p({\mathcal {V}}))}$. This implies that $\mu \in \partial \sigma (M_\varphi , l_p({\mathcal {V}}))$. Thus, there exists $\lambda \in \sigma (M_\varphi , l_p({\mathcal {V}}))$ such that $|\mu -\lambda |<\delta$. Hence, for all $i\in {{\mathbb {N}}}$ we get

$$\begin{aligned} |\varphi _i-\lambda |\ge |\varphi _i-\mu |-|\mu -\lambda |>\delta . \end{aligned}$$

Again, arguing as in the proof of Proposition 5.2, we deduce that $\lambda \in \rho (M_\varphi ,l_p({\mathcal {V}}))$, which is a contradiction. This prove that $\mu \notin \overline{\sigma (M_\varphi , l_p({\mathcal {V}}))}$ and hence $\overline{\sigma (M_\varphi , l_p({\mathcal {V}}))}\subseteq \overline{\{\varphi _i\,|\, i\in {{\mathbb {N}}}\}}$.

Now we prove the first equality of (5.2). It is sufficient to prove that if $\lambda \notin \overline{\{\varphi _i\,|\, i\in {{\mathbb {N}}}\}}$, then $\lambda \in \rho ^*(M_\varphi , l_p({\mathcal {V}}))$. So, fix $\lambda \notin \overline{\{\varphi _i\,|\, i\in {{\mathbb {N}}}\}}$. As done in the previous case, there exists $\delta >0$ such that if $|\mu -\lambda |<2\delta$, then $\mu \in \rho (M_\varphi ,l_p({\mathcal {V}}))$. Hence, we only have to show that the set $\{(M_\varphi -\mu I)^{-1}\,|\,|\mu -\lambda |<\delta \}$ is equicontinuous. By Lemma 5.1, we know that $\varphi _i\in \sigma (M_\varphi ,l_p({\mathcal {V}}))$, and thus $|\varphi _i-\mu |\ge 2\delta$ for all $i\in {{\mathbb {N}}}$. If $\mu \in {{\mathbb {C}}}$ is such that $|\mu -\lambda |<\delta$, then for all $i\in {{\mathbb {N}}}$ we get that $|\varphi _i-\mu |>\delta$. For $x\in l_p({\mathcal {V}})$ set $y^\mu := (M_\varphi -\mu I)^{-1}x$, i.e., $y^\mu =M_{\left( \frac{1}{\varphi _i-\mu }\right) _{i}}x$, for all $\mu \in {{\mathbb {C}}}$ with $|\mu -\lambda |<\delta$. Then, for all $i\in {{\mathbb {N}}}$ we have

$$\begin{aligned} |y_i^\mu |=\left| \frac{x_i}{\varphi _i-\mu }\right| \le \frac{|x_i|}{\delta }. \end{aligned}$$

Taking into account that $l_p({\mathcal {V}})=K_p({\overline{V}})$ topologically by assumption, the inductive limit topology is given by the seminorm system $(p_{{\overline{v}}})_{{\overline{v}}\in {\overline{V}}}$ and

$$\begin{aligned} p_{{\overline{v}}}\left( (M_\varphi -\mu I)^{-1}x \right) \le \frac{p_{{\overline{v}}}(x)}{\delta }, \end{aligned}$$

for all ${\overline{v}}\in {\overline{V}}$ and $\mu \in {{\mathbb {C}}}$ with $|\mu -\lambda |<\delta$. This proves the equicontinuity of the set $\{(M_\varphi -\mu I)^{-1}\,|\,|\mu -\lambda |<\delta \}$ and our thesis. $\square$

6 Ergodic properties

An operator $T\in {{\mathcal {L}}}(X)$, with X a lcHs, is called power bounded if $\{T^n\}_{n\in {{\mathbb {N}}}}$ is an equicontinuous subset of ${{\mathcal {L}}}(X)$.

The Cesàro means of an operator $T\in {{\mathcal {L}}}(X)$ are defined by

$$\begin{aligned} T_{[n]} :=\frac{1}{n}\sum _{m=1}^nT^m,\quad n\in {{\mathbb {N}}}. \end{aligned}$$

The operator T is called mean ergodic (resp. uniformly mean ergodic) if $\{T_{[n]}\}_{n\in {{\mathbb {N}}}}$ is a convergent sequence in ${{\mathcal {L}}}_s(X)$ (resp. in ${{\mathcal {L}}}_b(X)$). The Cesàro means of T satisfy the following identity

$$\begin{aligned} \frac{T^k}{k}= T_{[k]}-\frac{k-1}{k} T_{[k-1]}, \quad k \ge 2. \end{aligned}$$

So, it is clear that $\frac{T^k}{k}\rightarrow 0$ in ${{\mathcal {L}}}_s(X)$ as $k\rightarrow \infty$, whenever T is mean ergodic. Obviously, this holds also whenever T is power bounded. If X is a Montel lcHs, then the operator T is uniformly mean ergodic whenever it is mean ergodic. Furthermore, in reflexive Fréchet spaces (or (LF)-spaces, see [1, Corollary 2.7]) every power bounded operator is necessarily mean ergodic. If the space (or the (LF)-space, see [1, Proposition 2.8]) is Montel, every power bounded operator is necessarily uniformly mean ergodic. The converse of these two statements is not true in general (see, e.g., [15, Sect. 6]).

We start giving the following result. For Köthe echelon spaces, it is well-known (see [22, Lemma 3]).

Lemma 6.1

Let ${\mathcal {V}}$ be a system of weights and fix $\varphi =(\varphi _i)_{i\in {{\mathbb {N}}}}\in \omega$. For $1\le p\le \infty \, \cup \{0\}$, suppose that $T:=M_{\varphi }\in {{\mathcal {L}}}(l_p({\mathcal {V}}))$ satisfies that $\frac{T^k x}{k}\rightarrow 0$ as $k\rightarrow \infty$ for each $x\in l_p({\mathcal {V}})$. Then $\Vert \varphi \Vert _\infty \le 1$. This holds in particular if T is power bounded or mean ergodic.

Proof

For all $j\in {{\mathbb {N}}}$, let $e_j=(\delta _{i,j})_{i\in {{\mathbb {N}}}}$. The sequence $(e_j)_{j\in {{\mathbb {N}}}}$ clearly belongs to $l_p(V)$. By assumption,

$$\begin{aligned} \lim _{k\rightarrow \infty } \frac{|\varphi ^k_j|}{k}=\lim _{k\rightarrow \infty } \frac{|(T^ke_{j})_j|}{k}=0, \end{aligned}$$

with $(T^ke_{j})_j$ the j-th coordinate of $T^ke_j$. This implies $|\varphi _j|\le 1$ and so we get the thesis.

If T is power bounded or mean ergodic, then $\frac{T^k}{k}$ converges to 0 in ${{\mathcal {L}}}_s(l_p(V))$. $\square$

The condition of Lemma 6.1 is not only necessary, as the following theorem shows. We use the characterization of the power boundedness of the diagonal operators acting between Köthe echelon spaces, contained in [22, Proposition 2].

Lemma 6.2

Let $A=(a_n)_{n\in {{\mathbb {N}}}}$ be a Köthe matrix and fix $\varphi =(\varphi _i)_{i\in {{\mathbb {N}}}}\in \omega$. For $1\le p\le \infty \, \cup \{0\}$, the diagonal operator $M_{\varphi }\in {{\mathcal {L}}}(\lambda _p(A))$ is power bounded if, and only if, $\Vert \varphi \Vert _\infty \le 1$.

Theorem 6.3

Let ${\mathcal {V}}$ be a system of weights and fix $\varphi =(\varphi _i)_{i\in {{\mathbb {N}}}}\in \omega$. For $1\le p\le \infty \, \cup \{0\}$, the diagonal operator $M_{\varphi }\in {{\mathcal {L}}}(l_p({\mathcal {V}}))$ is power bounded if, and only if, $\Vert \varphi \Vert _\infty \le 1$.

Proof

If $M_\varphi$ is power bounded, from Lemma 6.1 we get that $\Vert \varphi \Vert _\infty \le 1$.

Suppose now that $\Vert \varphi \Vert _\infty \le 1$. For any $x\in l_p({\mathcal {V}})$, there exists $n\in {{\mathbb {N}}}$ such that $x\in \lambda _p(V_n)$. It is easy to see that under the assumption $|\varphi _i|\le \Vert \varphi \Vert _\infty \le 1$ for all $i\in {{\mathbb {N}}}$, the diagonal operator is defined pointwise in such a way: $M_\varphi :\lambda _p(V_n)\rightarrow \lambda _p(V_n)$. Applying Lemma 6.2, we get that the diagonal operator is power bounded on the step $\lambda _p(V_n)$. This implies that it is power bounded on $l_p({\mathcal {V}})$. $\square$

The same also holds for the mean ergodicity. Again, we need to recall the characterization of the mean ergodicity for the diagonal operators acting between Köthe echelon spaces given in [22, Theorem 2].

Lemma 6.4

Let $A=(a_n)_{n\in {{\mathbb {N}}}}$ be a Köthe matrix and fix $\varphi =(\varphi _i)_{i\in {{\mathbb {N}}}}\in \omega$. For $1\le p< \infty \cup \{0\}$, the diagonal operator $M_{\varphi }\in {{\mathcal {L}}}(\lambda _p(A))$ is mean ergodic if, and only if, $\Vert \varphi \Vert _\infty \le 1$.

Theorem 6.5

Let ${\mathcal {V}}$ be a system of weights and fix $\varphi =(\varphi _i)_{i\in {{\mathbb {N}}}}\in \omega$. For $1\le p< \infty \, \cup \{0\}$, the diagonal operator $M_{\varphi }\in {{\mathcal {L}}}(l_p({\mathcal {V}}))$ is mean ergodic if, and only if, $\Vert \varphi \Vert _\infty \le 1$.

Proof

If $M_\varphi$ is mean ergodic, from Lemma 6.1 we get that $\Vert \varphi \Vert _\infty \le 1$.

Suppose now that $\Vert \varphi \Vert _\infty \le 1$. As done in Theorem 6.3, fixed $x\in l_p({\mathcal {V}})$, the diagonal operator is defined pointwise from $\lambda _p(V_n)$ to $\lambda _p(V_n)$, for some $n\in {{\mathbb {N}}}$. Applying Lemma 6.4, we get that $\{(M_\varphi )_{[k]}x\}_{k\in {{\mathbb {N}}}}$ is a convergent sequence in $\lambda _p(V_n)$. Since $\lambda _p(V_n)$ is continuously embedded into $l_p({\mathcal {V}})$, we obtain the thesis. $\square$

Observe that in Theorems 6.3 and 6.5 completeness of $l_p({\mathcal {V}})$ was not required.

Finally, we discuss the uniform mean ergodicity. In contrast to what happens for the diagonal operators acting between Köthe echelon spaces [22, Theorem 3], for the sequence (LF)-spaces $l_p({\mathcal {V}})$ we have to require the completeness of the inductive limit.

Theorem 6.6

Let ${\mathcal {V}}$ be a system of weights and fix $\varphi =(\varphi _i)_{i\in {{\mathbb {N}}}}\in \omega$. For all $1\le p\le \infty \, \cup \{0\}$, suppose that $l_p({\mathcal {V}})=K_p({\overline{V}})$ algebraically and topologically (see (3.2), (3.3), (3.4)). The following assertions are equivalent:

• (1) $M_\varphi \in {{\mathcal {L}}}(l_\infty ({\mathcal {V}}))$ is mean ergodic;

• (2) $M_\varphi \in {{\mathcal {L}}}(l_\infty ({\mathcal {V}}))$ is uniformly mean ergodic;

• (3) $M_\varphi \in {{\mathcal {L}}}(l_0({\mathcal {V}}))$ is uniformly mean ergodic;

• (4) For $1\le p <\infty$, $M_\varphi \in {{\mathcal {L}}}(l_p({\mathcal {V}}))$ is uniformly mean ergodic;

• (5) $\Vert \varphi \Vert _\infty \le 1$ and for all $n\in {{\mathbb {N}}}$, for each ${\overline{v}}_n\in \lambda _\infty (V_n)_+$ and ${\overline{v}}\in {\overline{V}}$

  $$\begin{aligned} \lim _{k\rightarrow \infty } \sup _{i\in {{\mathbb {N}}}\setminus J} \frac{{\overline{v}}_i{\overline{v}}_n(i) |\varphi _i||1-\varphi _i^k|}{k\,|1-\varphi _i|}=0, \end{aligned}$$

  where $J:=\{i\in {{\mathbb {N}}}\,:\,\varphi _i=1 \}$.

Proof

Fix a seminorm $p_{{\overline{v}}}$. In the present case these seminorms define the lc-topology of $l_p({\mathcal {V}})$. We may assume without loss of generality that $\varphi _i \ne 1$ for all $i \in {{\mathbb {N}}}$, that means $J=\emptyset$. Otherwise, we split the space into two sectional subspaces and observe that in the subspace in which $\varphi _i = 1$, the diagonal operator acts as the identity.

(1)$\Rightarrow$(5). Clearly $\Vert \varphi \Vert _\infty \le 1$, by Lemma 6.1. Fix $n\in {{\mathbb {N}}}$, ${\overline{v}}\in {\overline{V}}$ and ${\overline{v}}_n\in \lambda _\infty (V_n)_+$. Since $M_\varphi$ is mean ergodic and ${\overline{v}}_n\in \lambda _\infty (V_n)_+\subset l_\infty ({\mathcal {V}})$, we have

$$\begin{aligned}&\lim _{k\rightarrow \infty } \sup _{i\in {{\mathbb {N}}}} \frac{{\overline{v}}_i{\overline{v}}_n(i) |\varphi _i||1-\varphi _i^k|}{k\,|1-\varphi _i|}=\lim _{k\rightarrow \infty }p_{{\overline{v}}}\left( (M_\varphi )_{[k]}{\overline{v}}_n\right) =0. \end{aligned}$$

(5)$\Rightarrow$(4). We show that for a fixed $B\in {{\mathcal {B}}}(l_p({\mathcal {V}}))$

$$\begin{aligned} \sup _{(x_i)_i\in B} p_{{\overline{v}}}\left( (M_\varphi )_{[k]}(x)\right) \rightarrow 0 \end{aligned}$$

as $k\rightarrow \infty$. Observe that

$$\begin{aligned} \sup _{(x_i)_i\in B} p_{{\overline{v}}}\left( (M_\varphi )_{[k]}(x)\right) =\sup _{(x_i)_i\in B} \left( \sum _{i\in {{\mathbb {N}}}}\left( \frac{{\overline{v}}_i|x_i| |\varphi _i||1-\varphi _i^k|}{k\,|1-\varphi _i|}\right) ^p\right) ^{\frac{1}{p}}. \end{aligned}$$

Taking into account that by assumption $l_p({\mathcal {V}})$ is regular (Theorem 3.4), there exists $n\in {{\mathbb {N}}}$ such that $B\in {{\mathcal {B}}}(\lambda _p(V_n))$. Hence, we can choose ${\overline{v}}_n\in \lambda _\infty (V_n)_+$ as in Proposition 4.9, getting

$$\begin{aligned} \sup _{(x_i)_i\in B} p_{{\overline{v}}}\left( (M_\varphi )_{[k]}(x)\right)&\le \sup _{i\in {{\mathbb {N}}}}\frac{{\overline{v}}_i{\overline{v}}_n(i) |\varphi _i||1-\varphi _i^k|}{k\,|1-\varphi _i|} \sup _{(x_i)_i\in B} \left( \sum _{i\in {{\mathbb {N}}}}\left( \frac{|x_i| }{{\overline{v}}_n(i)}\right) ^p\right) ^{\frac{1}{p}}\\ {}&\le \sup _{i\in {{\mathbb {N}}}}\frac{{\overline{v}}_i{\overline{v}}_n(i) |\varphi _i||1-\varphi _i^k|}{k\,|1-\varphi _i|}\rightarrow 0,\quad \text {as}\,k\rightarrow \infty , \end{aligned}$$

by assumption.

(5)$\Rightarrow$(2). It follows as in the previous case arguing with the $l_\infty$ norm.

(2)$\Rightarrow$(1). Trivial.

(4)$\Rightarrow$(5). Suppose that $M_\varphi$ is uniformly mean ergodic. In particular, $M_\varphi$ is mean ergodic, hence from Theorem 6.5$\Vert \varphi \Vert _\infty \le 1$. Fix ${\overline{v}}_n\in \lambda _\infty (V_n)_+$ and ${\overline{v}}\in {\overline{V}}$ and set $B:=\left\{ x\in \omega \,|\,\left\| \left( \frac{x_i}{{\overline{v}}_n(i)}\right) _{i\in {{\mathbb {N}}}}\right\| _p\le 1 \right\}$. B is bounded in $\lambda _p(V_n)$ from Proposition 4.9 and hence in $l_p({\mathcal {V}})$. Let $x\in B$.

Given $j\in {{\mathbb {N}}}$ such that $\varphi _j\ne 1$, we set $y^j_j:={\overline{v}}_n(j)$ and $y^j_i:=0$ if $j\ne i$. Then we put $y^j=(y_i^j)_{i\in {{\mathbb {N}}}}$. We have that $y^j\in B$ and

$$\begin{aligned} p_{{\overline{v}}}\left( (M_\varphi )_{[k]}(y^j)\right) =\frac{{\overline{v}}_j{\overline{v}}_n(j) |\varphi _j||1-\varphi _j^k|}{k\,|1-\varphi _j|}. \end{aligned}$$

Therefore

$$\begin{aligned} \sup _{j\in {{\mathbb {N}}}, \varphi _j\ne 1} \frac{{\overline{v}}_j{\overline{v}}_n(j) |\varphi _j||1-\varphi _j^k|}{k\,|1-\varphi _j|}\le \sup _{(x_i)_i\in B} p_{{\overline{v}}}\left( (M_\varphi )_{[k]}(x)\right) \rightarrow 0, \quad \text {as}\, k\rightarrow \infty \end{aligned}$$

since $M_\varphi$ is uniformly mean ergodic.

(3)$\Rightarrow$(5). It follows as in the previous case arguing with the $l_\infty$ norm.

(2)$\Rightarrow$(3). Trivial. $\square$