Spectra and dynamics of generalized Cesàro operators in (LF) and (PLB) sequence spaces

Abstract

In this paper, we introduce inductive limits of the Fréchet spaces $\ell (p+)$, $\text {ces}(p+)$, and $d(p+)$ ($1 \le p < \infty$) and projective limits of the (LB)-spaces $\ell (p-)$, $\text {ces}(p-)$, and $d(p-)$ ($1 < p \le \infty$). After having established some topological properties of such spaces as acyclicity and ultrabornologicity, we prove that the generalized Cesàro operators $C_t$ ($0 \le t \le 1$) act continuously in these sequence spaces, and we determine the spectra. Finally, we study the ergodic properties, that is, power boundedness, (uniform) mean ergodicity, and supercyclicity, of the operators $C_t$ acting in the (LF)-spaces and in the (PLB)-spaces mentioned above.

1 Introduction

Let $\omega :={\mathbb {C}}^{{\mathbb {N}}_0}$ and $0 \le t \le 1$. The discrete generalized Cesàro operator $C_t:\omega \rightarrow \omega$ is given by

$$\begin{aligned} C_t(x):= \Big ( \frac{t^n x_0 + t^{n-1}x_1 + \cdots + x_n}{n+1} \Big )_{n \in {\mathbb {N}}_0}, \qquad x=(x_n)_{n\in {\mathbb {N}}_0} \in \omega . \end{aligned}$$

(1.1)

For $t=1$, we recover the Cesàro operator $C_1$, given by

$$\begin{aligned} C_1(x):= \Big ( \frac{x_0 + x_1 + \cdots + x_n}{n+1} \Big )_{n \in {\mathbb {N}}_0}, \qquad x=(x_n)_n \in \omega . \end{aligned}$$

(1.2)

The spectra of the Cesàro operator $C_1$ defined in (1.2) have been analyzed in several Banach sequence spaces. For instance, in $\ell _p$, for $1<p<\infty$, [17, 18, 24, 31], in the discrete Cesàro space $\text {ces}(p)$ [19] and in d(p) [14], for $1<p<\infty$, which is isomorphic to the strong dual of $\text {ces}(p')$, with $p'$ the conjugate exponent of p, [12]. We refer the reader to [8] for a vast list of references concerning the study of the Cesàro operator $C_1$ in other Banach sequence spaces. The research of the operators defined in (1.1) started later, in the 1980s. This study has been focused on the spectrum of the generalized Cesàro operator $C_t$ ($0 \le t < 1$) in $\ell _p$, for $1<p<\infty$, [38, 44], in $\text {ces}(p)$ [19], and in d(p) [14], for $1<p<\infty$, among others, showing, surprisingly enough, a behaviour rather different from that of the Cesàro operator $C_1$. See also [20, 39, 45]. All the spaces considered satisfy that they are continuously included in the Fréchet space $\omega$, when it is endowed with the coordinatewise convergence topology.

Since the inclusions $\ell _q \subseteq \ell _p$, $\text {ces}(q) \subseteq \text {ces}(p)$, and $d(q) \subseteq d(p)$ are continuous if $1<q\le p<\infty$, for given $1\le p<\infty$ and $\{p_n\}_{n \in {\mathbb {N}}_0}$ a sequence satisfying $p<p_{n+1}<p_n$, for $n\in {\mathbb {N}}_0$, with $p_n \downarrow p$, we can define the Fréchet spaces

$$\begin{aligned} \ell (p+) = \bigcap _{n \in {\mathbb {N}}_0} \ell _{p_n}, \quad \text {ces}(p+) = \bigcap _{n \in {\mathbb {N}}_0} \text {ces}(p_n), \quad d(p+) = \bigcap _{n \in {\mathbb {N}}_0} d(p_n).\qquad \end{aligned}$$

(1.3)

Besides, for given $1<p\le \infty$ and $\{p_n\}_{n \in {\mathbb {N}}_0}$ a sequence satisfying $p_n< p_{n+1} < p$, for $n\in {\mathbb {N}}_0$, with $p_n \uparrow p$, we can define the (LB)-spaces

$$\begin{aligned} \ell (p-) = \bigcup _{n \in {\mathbb {N}}_0} \ell _{p_n}, \quad \text {ces}(p-) = \bigcup _{n \in {\mathbb {N}}_0} \text {ces}(p_n), \quad d(p-) = \bigcup _{n \in {\mathbb {N}}_0} d(p_n).\qquad \end{aligned}$$

(1.4)

Recently, the spectra of the generalized Cesàro operators $C_t$ ($0 \le t < 1$) acting in these Fréchet and (LB)-spaces have been studied by the first author, Bonet, and Ricker in [8]. On the other hand, the same authors described the spectra of the Cesàro operator $C_1$ acting in the spaces mentioned above [2, 4, 6,7,8, 15, 16], whose behaviour is still diverse from that of the generalized Cesàro operators $C_t$ for $t\ne 1$.

The dynamics of the generalized Cesàro operators ($0\le t < 1$) acting in the Banach sequence spaces $\ell _p$, $\text {ces}(p)$, and d(p) ($1<p<\infty$), in the Fréchet spaces defined in (1.3), in the (LB)-spaces defined in (1.4), and in $\omega$, are discussed in [8], showing that such operators $C_t$ are power bounded and uniformly mean ergodic, but not supercyclic. This behaviour holds true for the Cesàro operator $C_1$ in $\omega$ [2, 7]. However, this contrasts with the fact that $C_1$ is not power bounded nor mean ergodic nor supercyclic in the aforementioned Banach spaces [4, 5, 11, 14], Fréchet spaces [2, 7, 16] and (LB)-spaces [6, 8, 16].

The aim of this paper is twofold: to introduce and analyze the inductive limits of the Fréchet spaces $\ell (p+)$, $\text {ces}(p+)$, and $d(p+)$ ($1 \le p < \infty$), and the projective limits of the (LB)-spaces $\ell (p-)$, $\text {ces}(p-)$, and $d(p-)$ ($1 < p \le \infty$), and to study the spectra and the dynamics of the generalized Cesàro operators $C_t$ (1.1), $0\le t<1$, acting in these new sequence spaces, comparing them to those of the generalized Cesàro operator $C_t$ acting in the Fréchet and (LB)-spaces, and making the appropriate rearrangements for the case of the Cesàro operator $C_1$ (1.2).

The paper is divided as follows: in Sect. 2, we introduce some notation and present preliminary results on general (LF) and (PLB)-spaces. We also recall some properties regarding the Fréchet and (LB)-spaces mentioned above (see (1.3) and (1.4)).

In Sect. 3 we define the (LF)-spaces $L(p-)$, $C(p-)$, $D(p-)$, with $1<p\le \infty$ (the (PLB)-spaces $L(p+)$, $C(p+)$, $D(p+)$, with $1\le p<\infty$) as inductive limits of the Fréchet spaces given in (1.3) (as projective limits of the (LB)-spaces given in (1.4)), and study their topological properties such as acyclicity, completeness, reflexitivity or Montel (such as bornologicity, barrelledness, reflexitivity or Montel), see Proposition 2, Corollary 3 and Proposition 4. In Sect. 3 we also establish that $L(p-)\subseteq L(q-)$, $C(p-)\subseteq C(q-)$, $D(p-)\subseteq D(q-)$ ($L(p+)\subseteq L(q+)$, $C(p+)\subseteq C(q+)$, $D(p+)\subseteq D(q+)$), with continuous inclusions, for $1<p\le q$ (for $1\le p\le q$), as in the (LB)-space case (as in the Fréchet-space case), see Propositions 7 and 9.

In Sect. 4 we determine the spectra of generalized Cesàro operators $C_t$ ($0 \le t \le 1$) acting in the (LF) and (PLB)-spaces introduced in Sect. 3 (see Theorems 15, 20, and 22). The proofs of the results are based on Lemmas 10 and 12. In particular, we show that the spectra of $C_t$ ($0 \le t <1$) acting in these (LF) and (PLB)-spaces coincide with those in the corresponding Fréchet and (LB)-spaces (Theorem 15). We obtain similar results also for the spectra of $C_1$ acting in these (LF) and (PLB)-spaces (Theorems 20 and 22).

Finally, in Sect. 5 we study the ergodic properties, i.e., the power boundedness, the mean ergodicity, and the uniform mean ergodicity, of continuous linear operators acting in (LF)-spaces (in (PLB)-spaces). We compare such properties with the same of the continuous linear operators acting in the steps of their inductive spectrum (projective spectrum), see Theorems 24 and 26. We then apply Theorems 24 and 26 to establish that the generalized Cesàro operators $C_t$ ($0\le t<1$) acting in the (LF) and (PLB)-spaces introduced in Sect. 3 are power bounded, mean ergodic, and uniformly mean ergodic (Corollaries 25 and 27). We conclude showing that the Cesàro operator $C_1$ acting in these sequence spaces is, in contrast, not power bounded nor mean ergodic, and that $C_t$ is never supercyclic for every $0\le t\le 1$.

2 Notation and preliminary results

For two locally convex Hausdorff spaces (lcHs, for short) X and Y, we denote by ${\mathcal {L}}(X,Y)$ the space of all continuous linear operators $T:X\rightarrow Y$. We write ${\mathcal {L}}(X) = {\mathcal {L}}(X,X)$ and we denote by $\Gamma = \Gamma _X$ a collection of continuous seminorms that determine the topology of X. The identity operator on ${\mathcal {L}}(X)$ is denoted by I and the set of bounded sets of X is denoted by ${\mathcal {B}}(X)$. When the space ${\mathcal {L}}(X)$ is endowed with the strong operator topology (the topology of uniform convergence on ${\mathcal {B}}(X)$), it is denoted by ${\mathcal {L}}_s(X)$ (by ${\mathcal {L}}_b(X)$). The topological dual of X is denoted by $X':= {\mathcal {L}}(X, {\mathbb {C}})$. We write $X'_{\sigma }$ (we write $X'_{\beta }$) for the space $X'$ endowed with the weak* topology $\sigma (X',X)$ (the strong topology $\beta (X',X)$). Some standard references of functional analysis are [27, 30, 34, 42].

Let X be a lcHs. We will say that X is an (LF)-space if there exists an increasing sequence $\{ X_n \}_{n \in {\mathbb {N}}}$ of Fréchet spaces such that the inclusion $X_n \subseteq X_{n+1}$, for $n\in {\mathbb {N}}$, is continuous and the topology in $X = \bigcup _n X_n$ coincides with the finest locally convex topology for which each inclusion $X_n \subseteq X$ is continuous. We will denote it by $X = \text {ind}_n X_n$; the sequence $\{ X_n \}_{n\in {\mathbb {N}}}$ is called a defining inductive spectrum of X. We will say that X is an (LB)-space if each $X_n$ is a Banach space. We remark that every (LF)-space is ultrabornological [34, Remark 24.36], hence bornological and barrelled.

An (LF)-space $X = \text {ind}_n X_n$ is said to be regular if every $B \in {\mathcal {B}}(X)$ is contained and bounded in $X_n$, for some n. Every complete (LF)-space is regular. We will say that X satisfies condition (M) (condition $(M_0)$) of Retakh [37] if there exists an increasing sequence $\{ U_n \}_{n \in {\mathbb {N}}} \subseteq X$ such that $U_n$ is an absolutely convex zero-neighbourhood of $(X_n,\tau _n)$, $n \in {\mathbb {N}}$, for which

• (M) for every n there exists $m\ge n$ such that for every $l \ge m$, the topologies $\tau _l$ and $\tau _m$ induce the same topology on $U_n$.

• (($M_0$) for every n there exists $m\ge n$ such that for every $l \ge m$, the topologies $\sigma (X_l, X'_l)$ and $\sigma (X_m, X'_m)$ induce the same topology on $U_n$.)

Recall that Vogt [41, Theorem 2.10] established that an (LF)-space X satisfies condition (M) (condition $(M_0)$) if, and only if, it is acyclic (weakly acyclic) in the sense of Palamodov [36]. Acyclic (LF)-spaces are complete [36, Corollary 7.1] (cf. [41, Theorem 3.2]). Moreover, a standard duality proof shows that (M) implies $(M_0)$.

If $\{ \left\Vert \cdot \right\Vert _{n,\ell } \}_{\ell \in {\mathbb {N}}}$ denotes a fundamental system of seminorms of the Fréchet space $X_n$, for $n \in {\mathbb {N}}$, we then say that the (LF)-space $X=\text {ind}_nX_n$ satisfies the condition (Q) (the condition (wQ)) if

• (Q) $\forall \! \ n \ \exists \! \ m>n, \ N \in {\mathbb {N}} \ \ \forall \! \ k>m, \ M \in {\mathbb {N}}, \ \varepsilon>0, \ \exists \ \! K \in {\mathbb {N}}, \ S>0 \ \forall x \in X_n$,

  $$\begin{aligned} \left\Vert x\right\Vert _{m,M} \le S\left\Vert x\right\Vert _{k,K} + \varepsilon \left\Vert x\right\Vert _{n,N}. \end{aligned}$$

• ((wQ) $\forall \! \ n \ \exists \! \ m>n, \ N \in {\mathbb {N}} \ \ \forall \! \ k>m, \ M \in {\mathbb {N}} \ \exists \ \! K \in {\mathbb {N}}, \ S>0 \ \forall x \in X_n$,

  $$\begin{aligned} \left\Vert x\right\Vert _{m,M} \le S(\left\Vert x\right\Vert _{k,K} + \left\Vert x\right\Vert _{n,N}). ) \end{aligned}$$

The condition (wQ) is clearly weaker than the condition (Q). The conditions (Q) and (wQ) were introduced and studied in [41]. In particular, in [41] it was shown that such conditions are necessary for the acyclicity and the weak acyclicity and, under further assumptions, also sufficient. Thereafter, Wengenroth [43, Theorems 2.7 and 3.3] proved that the conditions (M) and (Q) ((wQ) under suitable assumptions on the spaces $X_n$) are equivalent. More precisely,

Theorem 1

Let $X=\text {ind}_nX_n$ be an (LF)-space. The space X satisfies condition (M) (i.e., it is acyclic) if, and only if, X satisfies condition (Q). Furthermore, if X is an inductive limit of Fréchet-Montel spaces, then X satisfies condition (M) (i.e., it is acyclic) if, and only if, it is complete if, and only if, it satisfies (wQ).

We will say that a lcHs X is a (PLB)-space if there exists a decreasing sequence $\{ X_n \}_{n \in {\mathbb {N}}}$ of (LB)-spaces such that the inclusion $X_{n+1} \subseteq X_n$ is continuous, for $n\in {\mathbb {N}}$, and the topology in $X = \bigcap _n X_n$ coincides with the coarsest locally convex topology for which each inclusion $X \subseteq X_n$ is continuous. We will denote it by $X = \text {proj}_n X_n$; the sequence $\{ X_n \}_{n \in {\mathbb {N}}}$ is called a defining projective spectrum of X. It follows that a (PLB)-space X is complete provided $X_n$ is a complete (LB)-space for infinitely many n. The (PLB)-space X is called reduced if the inclusion $X\subseteq X_n$ has dense range for every $n\in {\mathbb {N}}$. The (PLB)-space X is a Fréchet space if each $X_n$ is a Banach space.

The main problem concerning (PLB)-spaces consists in discussing whether they are bornological/barrelled. In the case that $X=\text {proj}_n (X_n)'_\beta$ is a reduced (PLB)-space of strong duals of reflexive Fréchet spaces $X_n$, if the (LF)-space $\text {ind}_nX_n$ is weakly acyclic, then X is bornological, as shown by Vogt in [41, Lemma 4.2]. Under the assumption that the steps $X_n$ are Fréchet-Montel spaces, Wengenroth [43, Theorem 3.5] proved that $X=\text {proj}_n (X_n)'_\beta$ is bornological if, and only if, the (LF)-space $\text {ind}_nX_n$ satisfies the condition (wQ). We refer to [40, 41, 43] for further results.

For $1\le p\le \infty$, we write $\left\Vert \cdot \right\Vert _p$ for the standard norm in $\ell _p$. For $1<p<\infty$ we define

$$\begin{aligned} \text {ces}(p) = \{ x \in \omega : \, \ \left\Vert x\right\Vert _{\text {ces}(p)} = \left\Vert C_1(|x|)\right\Vert _p < \infty \}. \end{aligned}$$

The Banach spaces $\text {ces}(p)$, $1<p<\infty$, have been deeply studied in Bennett [12] (see also [10, 19, 25, 32]).

For $1<p<\infty$, the dual Banach spaces $(\text {ces}(p))'$ are rather complicated (see [29]). An isomorphic identification of $(\text {ces}(p))'$ is given in [12, Corollary 12.17], that is, it is shown there that

$$\begin{aligned} d(p) = \Big \{ x \in \ell _{\infty }: \hat{x} := \Big (\sup _{k\ge n} |x_k| \Big )_{n \in \mathbb {N}_0} \in \ell _p \Big \} \end{aligned}$$

is a Banach space, with the norm

$$\begin{aligned} \left\Vert x\right\Vert _{d(p)} = \left\Vert {\hat{x}}\right\Vert _p, \qquad x \in d(p), \end{aligned}$$

and that it is isomorphic to $(\text {ces}(p'))'$, where $1< p' < \infty$ satisfies $1/p + 1/p' = 1$.

For $1<p,q<\infty$, we have that $p \le q$ if, and only if, $\ell _p \subseteq \ell _q$, $\text {ces}(p) \subseteq \text {ces}(q)$ [5, Proposition 3.2(iii)], $d(p) \subseteq d(q)$ [14, Proposition 5.1(iii)], $\ell _p \subseteq \text {ces}(q)$ ([5, Proposition 3.2(ii)] and [19, Remark 2.2(ii)], where Hardy’s inequality [28, Theorem 326] is used), $d(p) \subseteq \ell _q$ [14, Proposition 2.7(v) and 5.1(ii)], and $d(p) \subseteq \text {ces}(q)$ [14, Proposition 5.1(i)] with continuous inclusions. We have that $\ell _p$, $\text {ces}(p)$ (and hence d(p)) ($1<p<\infty$) are reflexive ([12, p. 61] and [29, Proposition 2]), and also they are separable. They contain $\{e_n\}_{n\in {\mathbb {N}}_0}$ (here, $e_n=(\delta _{nk})_{k\in {\mathbb {N}}_0}$) as an unconditional basis. The spaces $\ell _p$, $\text {ces}(q)$, and d(r) are not isomorphic for $1< p,q, r < \infty$ (see [15, Proposition 2.2] for references).

We now recall some properties of $\omega$, of the Fréchet spaces defined in (1.3), and of the (LB)-spaces defined in (1.4). First of all, the definition of the Fréchet spaces above ((LB)-spaces above) is independent of the choice of the sequence $p_n \downarrow p$ ($p_n\uparrow p$). The spaces $\ell (p+)$ are studied in, among others, [21, 35]. The spaces $\ell (p+)$, for $1\le p<\infty$, are reflexive and separable, but not Montel. We refer to [4] (see also [7]) for the properties of the spaces $\text {ces}(p+)$. The spaces $\text {ces}(p+)$, for $1\le p<\infty$, are separable, Fréchet–Schwartz (hence Montel) [4, Theorem 3.1 and Corollary 3.2], but not nuclear [4, Proposition 3.5(ii)]. Clearly, $\ell _p \subseteq \ell (p+) \subseteq \omega$ and $\text {ces}(p) \subseteq \text {ces}(p+) \subseteq \omega$, for $1 \le p < \infty$, with continuous inclusions. We also know [4, Proposition 3.5(iii)] that $\ell (p+)$ is not isomorphic to any $\text {ces}(q+)$, $1\le q < \infty$. The spaces $d(p+)$, for $1\le p<\infty$, are also separable, Fréchet–Schwartz, not nuclear, and not isomorphic to any $\text {ces}(q+)$, $1 \le q < \infty$ [15, Theorem 4.7].

As in the Banach case, we have that $1 \le p \le q < \infty$ if, and only if, $\ell (p+) \subseteq \ell (q+)$ [7, Proposition 26(ii)], $\text {ces}(p+)\subseteq \text {ces}(q+)$ ( [7, Proposition 26(iii)]), $d(p+) \subseteq d(q+)$ (see for instance [15, (4.2)]), $\ell (p+) \subseteq \text {ces}(q+)$ [7, Proposition 26(ii)], $d(p+) \subseteq \ell (q+)$ and $d(p+) \subseteq \text {ces}(q+)$ (see for instance [15, (4.4)]), with continuous inclusions.

On the other hand, it is known (see again [21, 35]) that the space $\ell (p-)$, for $1<p\le \infty$, is complete, reflexive, but not Montel. Furthermore, $\ell (p-)$ is isomorphic to $(\ell (p'+))'_{\beta }$, and $(\ell (p-))'_{\beta }$ is isomorphic to $\ell (p'+)$, where $1 \le p' < \infty$ satisfies $1/p + 1/p' = 1$. In [6], the spaces $\text {ces}(p-)$ are studied. There, it is shown that $\text {ces}(p-)$ (and also $d(p-)$), with $1<p\le \infty$, are reflexive, separable, Montel (see also [13, pp. 61–62]) and not nuclear. Furthermore, by [6, Proposition 5.1], we have that the inclusions $\ell (p-) \subseteq \ell (q-)$, $\text {ces}(p-) \subseteq \text {ces}(q-)$, $d(p-) \subseteq d(q-)$, $\ell (p-) \subseteq \text {ces}(q-)$, $d(p-) \subseteq \ell (q-)$, $d(p-) \subseteq \text {ces}(q-)$ are continuous provided $1 < p \le q \le \infty$. By [15, Proposition 4.3 and Remark 4.4], it holds that

$$\begin{aligned} (d(p+))'_{\beta } = \text {ces}(p'-), \quad 1 \le p< \infty \qquad \text {and} \qquad (d(p-))'_{\beta } = \text {ces}(p'+), \quad 1 < p \le \infty \end{aligned}$$

isomorphically, where $p'$ satisfies $1/p+1/p'=1$. Therefore, the spaces $\text {ces}(p-)$ and $d(p-)$, for $1<p\le \infty$, are (DFS)-spaces.

3 (LF) and (PLB) sequence spaces

For the Fréchet spaces $\ell (p+)$, $\text {ces}(p+)$, $d(p+)$ given in (1.3), we have that $\ell (p+) \subseteq \ell (q+)$, $\text {ces}(p+) \subseteq \text {ces}(q+)$, and $d(p+) \subseteq d(q+)$ continuously if and only if $1 \le p \le q < \infty$ (the continuous inclusions have dense range because $\{e_n\}_{n\in \mathbb {N}_0}$ is a basis in each of the spaces $\ell (p+)$, $\text {ces}(p+)$, $d(p+)$). Thus, for given $1 < p \le \infty$ and $\{ p_n \}_n$ a sequence satisfying $p_n< p_{n+1} < p$, for $n\in {\mathbb {N}}_0$, with $p_n \uparrow p$, we can define the (LF)-spaces

$$\begin{aligned} L(p-):= \bigcup _{n \in {\mathbb {N}}_0} \ell (p_n+), \qquad C(p-):= \bigcup _{n \in {\mathbb {N}}_0} \text {ces}(p_n+), \qquad D(p-):= \bigcup _{n \in {\mathbb {N}}_0} d(p_n+). \end{aligned}$$

On the other hand, we also have that the (LB)-spaces given in (1.4) satisfy that $\ell (p-) \subseteq \ell (q-)$, $\text {ces}(p-) \subseteq \text {ces}(q-)$, and $d(p-) \subseteq d(q-)$ continuously, provided $1 < p \le q \le \infty$ (the continuous inclusions have dense range because $\{e_n\}_{n\in \mathbb {N}_0}$ is a basis in each of the spaces $\ell (p-)$, $\text {ces}(p-)$, $d(p-)$). Thus, for given $1 \le p < \infty$ and $\{ p_n \}_n$ a sequence satisfying $p< p_{n+1} < p_n$, for $n\in {\mathbb {N}}_0$, and $p_n \downarrow p$, we can define the (PLB)-spaces

$$\begin{aligned} L(p+):= \bigcap _{n \in {\mathbb {N}}_0} \ell (p_n-), \qquad C(p+):= \bigcap _{n \in {\mathbb {N}}_0} \text {ces}(p_n-), \qquad D(p+):= \bigcap _{n \in {\mathbb {N}}_0} d(p_n-). \end{aligned}$$

The definition of the (LF)-spaces above ((PLB)-spaces above) is independent of the choice of the sequence $p_n \uparrow p$ ($p_n\downarrow p$).

Clearly, the (LF)-spaces $L(p-)$, $C(p-)$ and $D(p-)$ are continuously included in $\omega$, as well as the (PLB)-spaces $L(p+)$, $C(p+)$ and $D(p+)$.

Proposition 2

For $1<p\le \infty$, the spaces $L(p-)$, $C(p-)$, and $D(p-)$ are acyclic and hence, complete.

Proof

By Theorem 1, it is enough to show that these spaces satisfy the condition (Q). To this end, we will use a well-known interpolation estimate: for $1<p<q<r$,

$$\begin{aligned} \left\Vert \cdot \right\Vert _q \le \left\Vert \cdot \right\Vert _r^{\theta } \left\Vert \cdot \right\Vert _{p}^{1-\theta }, \end{aligned}$$

(3.1)

with $\theta = \frac{r(q-p)}{q(r-p)} \in (0,1)$, where $\left\Vert \cdot \right\Vert _s$ denotes the $\ell _s$-norm (see, f.i., [33, Proposition 1.d.2(ii). p.43]). In order to apply (3.1), we observe that for every $\theta \in (0,1)$ and $x,y\ge 0$ we have

$$\begin{aligned} x^{\theta } y^{1-\theta } \le x+y, \end{aligned}$$

as it is easy to verify. Furthermore, we have for every $x,y\ge 0$ and $\varepsilon >0$ that

$$\begin{aligned} x^{\theta } y^{1-\theta } = \frac{1}{\varepsilon ^{1-\theta }} x^{\theta } \varepsilon ^{1-\theta } y^{1-\theta } = \Big ( \frac{1}{\varepsilon ^{(1-\theta )/\theta }} x\Big )^{\theta } \cdot (\varepsilon y)^{1-\theta }. \end{aligned}$$

Therefore, we deduce for every $\varepsilon >0$, $\theta \in (0,1)$ and $x,y \ge 0$ that

$$\begin{aligned} x^{\theta } y^{1-\theta } \le \frac{1}{\varepsilon ^{(1-\theta )/\theta }}x + \varepsilon y. \end{aligned}$$

Thus, by (3.1) it follows for every $1<p<q<r$ and $\varepsilon >0$ that

$$\begin{aligned} \left\Vert \cdot \right\Vert _q \le \frac{1}{\varepsilon ^{(1-\theta )/\theta }} \left\Vert \cdot \right\Vert _r + \varepsilon \left\Vert \cdot \right\Vert _ p. \end{aligned}$$

(3.2)

Now, for a fixed $1<p\le \infty$, let $\{ p_n \}_{n\in {\mathbb {N}}}$ be any strictly increasing sequence satisfying $p_n \uparrow p$. Then for any $n \in {\mathbb {N}}$, we choose a strictly decreasing sequence $\{ p_{n,{l}} \}_{l\in {\mathbb {N}}}$ satisfying $p_n< p_{n,l} < p_{n+1}$, for $l\in {\mathbb {N}}$, and $p_{n, l} \downarrow p_n$, and denote by $\Vert \cdot \Vert _{n,l}$ the $\ell _{p_{n,l}}$-norm for every $l\in {\mathbb {N}}$. Then $\{\Vert \cdot \Vert _{n,l}\}_{l\in {\mathbb {N}}}$ is a fundamental system of seminorms of the Fréchet space $\ell (p_n+)$ for every $n\in {\mathbb {N}}$.

By (3.2), for every $n \in {\mathbb {N}}$ there exists $m=n+1$ such that for every $k>m$, $N,M,K\in {\mathbb {N}}$ and $\varepsilon >0$ there exists $S=\frac{1}{\varepsilon ^{(1-\theta )/\theta }}>0$, where $\theta =\frac{p_{k,K}(p_{n+1,M}-p_{n,N})}{p_{n+1,M}(p_{k,K}-p_{n,N})}$, such that

$$\begin{aligned} \left\Vert x\right\Vert _{m,M} \le S\left\Vert x\right\Vert _{k,K} + \varepsilon \left\Vert x\right\Vert _{n,N}, \qquad \forall x\in \ell (p_n+). \end{aligned}$$

(3.3)

This implies that $L(p-)$ satisfies the condition (Q). So, by Theorem 1, the (LF)-space $L(p-)$ satisfies condition (M), i.e., it is acyclic and hence, complete.

To show the acyclicity of the (LF)-space $C(p-)$ ($D(p-)$), it suffices to replace x in (3.3) by $C_1(|x|)$ (by ${\hat{x}}$). Indeed, proceeding in this way, we obtain that for every $n \in {\mathbb {N}}$ there exists $m=n+1$ such that for every $k>m$, $N,M,K\in {\mathbb {N}}$ and $\varepsilon >0$ there exists $S=\frac{1}{\varepsilon ^{(1-\theta )/\theta }}>0$, where $\theta =\frac{p_{k,K}(p_{n+1,M}-p_{n,N})}{p_{n+1,M}(p_{k,K}-p_{n,N})}$, such that

$$\begin{aligned} \left\Vert C_1(|x|)\right\Vert _{m,M}{} & {} \le S\left\Vert C_1(|x|)\right\Vert _{k,K} + \varepsilon \left\Vert C_1(|x|)\right\Vert _{n,N}, \qquad \forall x\in \text {ces}(p_n+) \\{} & {} \left( \left\Vert {\hat{x}}\right\Vert _{m,M} \le S\left\Vert {\hat{x}}\right\Vert _{k,K} + \varepsilon \left\Vert {\hat{x}}\right\Vert _{n,N}, \qquad \forall x \in d(p_n+)\right) , \end{aligned}$$

that is,

$$\begin{aligned} \left\Vert x\right\Vert _{\text {ces}(p_{m,M})}{} & {} \le S\left\Vert x\right\Vert _{\text {ces}(p_{k,K})} + \varepsilon \left\Vert x\right\Vert _{\text {ces}(p_{n,N})}, \qquad \forall x \in \text {ces}(p_n+) \\{} & {} \left( \left\Vert x\right\Vert _{d(p_{m,M})} \le S\left\Vert x\right\Vert _{d(p_{k,K})} + \varepsilon \left\Vert x\right\Vert _{d(p_{n,N})}, \qquad \forall x \in d(p_n+) \right) . \end{aligned}$$

Since $\{\left\Vert \cdot \right\Vert _{\text {ces}(p_{n,l})}\}_{l\in {\mathbb {N}}}$($\{\left\Vert \cdot \right\Vert _{d(p_{n,l})}\}_{l\in {\mathbb {N}}}$) is a fundamental system of seminorms of the Fréchet space $\text {ces}(p_n+)$ ($d(p_n+)$) for every $n\in {\mathbb {N}}$, we deduce that the (LF)-space $C(p-)$ ($D(p-))$) satisfies the condition (Q) and hence, by Theorem 1 it is acyclic and necessarily complete. $\square$

Proposition 2 clearly implies that

Corollary 3

Let $1<p\le \infty$. Then the following properties are satisfied.

• (i) The (LF)-space $L(p-)$ is reflexive.

• (ii) The (LF)-spaces $C(p-)$ and $D(p-)$ are Montel and hence, reflexive.

Proof

• (i) Since $L(p-)$ is an (LF)-space, it is clearly ultrabornological and hence, bornological and barrelled. So, to conclude that $L(p-)$ is reflexive, it suffices to show that the sets $B\in {\mathcal {B}}(L(p-))$ are relatively $\sigma (L(p-),(L(p-))')$-compact. So, we fix $B\in {\mathcal {B}}(L(p-))$. Since by Proposition 2 the (LF)-space $L(p-)$ is complete and hence regular, there exists $n\in {\mathbb {N}}$ such that B is contained and bounded in $\ell (p_n+)$. But, $\ell (p_n+)$ is a reflexive Fréchet space and so, B is relatively $\sigma (\ell (p_n+),(\ell (p_n+))')$-compact. This implies that B is necessarily relatively $\sigma (L(p-),(L(p-))')$-compact.

• (ii) The (LF)-space $C(p-)$ ($D(p-)$) is ultrabornological and hence, bornological and barrelled. So, to conclude that $C(p-)$ ($D(p-)$) is Montel, it suffices to show that the sets $B\in {\mathcal {B}}(C(p-))$ ($B\in {\mathcal {B}}(D(p-))$) are relatively compact. So, we fix $B\in {\mathcal {B}}(C(p-))$ ($B\in {\mathcal {B}}(D(p-))$). Since by Proposition 2 the (LF)-space $C(p-)$ ($D(p-)$) is complete and hence regular, there exists $n\in {\mathbb {N}}$ such that B is contained and bounded in $\text {ces}(p_n+)$ ($d(p_n+)$). But, by [4, Theorem 3.1 and Corollary 3.2] (by [15, Theorem 4.7]) $\text {ces}(p_n+)$ ($d(p_n+)$) is a Fréchet–Schwartz space and so, B is relatively compact in $\text {ces}(p_n+)$ ($d(p_n+)$). This implies that B is necessarily relatively compact in $C(p-)$ ($D(p-)$). $\square$

A further immediate consequence of Proposition 2 above combined with [41, Lemma 4.2] is the following result.

Proposition 4

Let $1\le p < \infty$. Then the following properties are satisfied.

• (i) The (PLB)-spaces $L(p+)$, $C(p+)$ and $D(p+)$ are bornological.

• (ii) The (PLB)-space $L(p+)$ is reflexive, whereas the (PLB)-spaces $C(p+)$ and $D(p+)$ are Montel.

Proof

(i) It suffices to give the proof only for the (PLB)-space $L(p+)$. The other cases follow in a similar way.

Let $1<p'\le \infty$ satisfy $1/p+1/p'=1$. Then $L(p'-)=\bigcup _{n\in {\mathbb {N}}}\ell (p'_n+)$, with $1<p'_n\uparrow p'$, is a reflexive, acyclic (hence, complete and weakly acyclic) (LF)-space (by Corollary 3 and Proposition 2). So, it follows that its strong dual $(L(p'-))'_\beta$ is canonically isomorphic to the projective limit of the strong duals of the spaces $\ell (p'_n+)$, i.e., $(L(p'-))'_\beta =\bigcap _{n\in {\mathbb {N}}} (\ell (p'_n+))'_\beta$ and that by [41, Lemma 4.2] the strong dual $(L(p'-))'_\beta$ is bornological. But, if for every $n\in {\mathbb {N}}$, we take $1\le p_n<\infty$ satisfying $1/p_n+1/p'_n=1$, then $p_n\downarrow p$ and $(\ell (p'_n+))'_\beta =\ell (p_n-)$. Therefore, we deduce that

$$\begin{aligned} L(p+)=(L(p'-))'_\beta \end{aligned}$$

and hence, it is bornological.

(ii) As it follows from the proof of point (i), we have that

$$\begin{aligned} L(p+)=(L(p'-))'_\beta , \quad C(p+)=(D(p'-))'_\beta , \quad D(p+)=(C(p'-))'_\beta , \end{aligned}$$

with $1< p' \le \infty$ satisfying $1/p+1/p'=1$. Accordingly, as $L(p+)$ ($C(p+)$ and $D(p+)$) is the strong dual of a reflexive (of a Montel) lcHs, it is reflexive (they are Montel). $\square$

As an immediate consequence of Corollary 3 and Proposition 4, we obtain that

Corollary 5

• (i) Let $1<p\le \infty$. If $1\le p'<\infty$ satisfies $1/p+1/p'=1$, then

  $$\begin{aligned} (L(p-))'_\beta = L(p'+), \quad (C(p-))'_\beta = D(p'+), \quad (D(p-))'_\beta = C(p'+). \end{aligned}$$

• (ii) Let $1\le p <\infty$. If $1<p'\le \infty$ satisfies $1/p+1/p'=1$, then

  $$\begin{aligned} (L(p+))'_\beta = L(p'-),\quad (C(p+))'_\beta = D(p'-), \quad (D(p+))'_\beta = C(p'-). \end{aligned}$$

We conclude this section with some results regarding the validity of some inclusions between the spaces introduced above. For this, we recall a characterization for the continuity of an operator in (LF)-spaces (see [27, p. 147]):

Lemma 6

Let $X=\text {ind}_n X_n$ and $Y=\text {ind}_n Y_n$ be two (LF)-spaces. Let $T:X\rightarrow Y$ be a linear operator. Then $T \in {\mathcal {L}}(X,Y)$ if, and only if, for each $n \in {\mathbb {N}}$ there exists $m\in {\mathbb {N}}$ such that $T(X_n) \subseteq Y_m$ and the restriction $T:X_n \rightarrow Y_m$ is continuous.

Applying Lemma 6 we immediately obtain that

Proposition 7

We have $L(p-) \subseteq L(q-)$, $C(p-) \subseteq C(q-)$, $D(p-) \subseteq D(q-)$, $L(p-) \subseteq C(q-)$, $D(p-) \subseteq L(q-)$, and $D(p-) \subseteq C(q-)$ with continuous inclusions if, and only if, $1 < p \le q \le \infty$.

Proof

It suffices to give the proof only for the first inclusion. The other cases follow in a similar way.

So, let $1<p< q < \infty$. If $1<p_n\uparrow p$ and $p\le q_n\uparrow q$, then $p_n<q_n$ for every $n\in {\mathbb {N}}$. Accordingly, $\ell (p_n+)\subseteq \ell (q_n+)$ with continuous inclusion for every $n\in {\mathbb {N}}$. Since $L(p-)=\bigcup _{n\in {\mathbb {N}}}\ell (p_n+)$ and $L(q-)=\bigcup _{n\in {\mathbb {N}}}\ell (q_n+)$, the continuity of the inclusion $L(p-) \subseteq L(q-)$ follows from Lemma 6.

Suppose that $L(p-)\subseteq L(q-)$ with continuous inclusion for some $1< p, q < \infty$. If $L(p-)=\bigcup _{n\in {\mathbb {N}}}\ell (p_n+)$ and $L(q-)=\bigcup _{n\in {\mathbb {N}}}\ell (q_n+)$ with $p_n\uparrow p$ and $q_n\uparrow q$ respectively, then by Lemma 6 for each $n\in {\mathbb {N}}$ there exists $m(n)\in {\mathbb {N}}$ such that $\ell (p_n+)\subseteq \ell (q_{m(n)}+)$ with continuous inclusion. But, for any $n\in {\mathbb {N}}$, $\ell (p_n+)\subseteq \ell (q_{m(n)}+)$ with continuous inclusion if, and only if, $p_n\le q_{m(n)}<q$. Letting $n\rightarrow \infty$, it follows that $p\le q$. $\square$

For analogous inclusions in the (PLB)-spaces considered, we state a characterization for the continuity in (PLB)-spaces given in [9, Proposition 2] (cf. [22, Lemma 4]).

Lemma 8

Let $X=\text {proj}_n X_n$ and let $Y=\text {proj}_n Y_n$ be (PLB)-spaces such that $X \subseteq X_n$ has dense range for all $n\in {\mathbb {N}}$, and each $Y_n$ is a complete (LB)-space. Let $T:X\rightarrow Y$ be a linear operator. We have that $T \in {\mathcal {L}}(X,Y)$ if and only if for all $n \in {\mathbb {N}}$ there exists $m \in {\mathbb {N}}$ such that T admits a unique continuous extension $T:X_m \rightarrow Y_n$.

Proposition 9

We have $L(p+) \subseteq L(q+)$, $C(p+) \subseteq C(q+)$, $D(p+) \subseteq D(q+)$, $L(p+) \subseteq C(q+)$, $D(p+) \subseteq L(q+)$, and $D(p+) \subseteq C(q+)$ with continuous inclusions if, and only if, $1\le p \le q < \infty$.

Proof

The proof is similar to the one of Proposition 7. Actually, it suffices to apply Lemma 8, after having observed that each (LB)-space in (1.4) is complete and that each (PLB)-space is dense in its steps as it contains the set $\{e_n\}_{n\in {\mathbb {N}}_0}$. $\square$

4 Spectra of generalized Cesàro operators in (LF) and (PLB)-spaces

Let X be a lcHs and $T \in {\mathcal {L}}(X)$. The resolvent set $\rho (T;X)$ of T consists of all $\lambda \in {\mathbb {C}}$ such that $R(\lambda , T):= (\lambda I - T)^{-1}$ exists in ${\mathcal {L}}(X)$. The spectrum of T is defined by $\sigma (T;X):= {\mathbb {C}} \setminus \rho (T;X)$. The point spectrum $\sigma _{pt}(T;X) \subseteq \sigma (T;X)$ consists of all $\lambda \in {\mathbb {C}}$ such that $(\lambda I - T)$ is not injective. The elements in $\sigma _{pt}(T;X)$ are called eigenvalues. An eigenvalue $\lambda \in {\mathbb {C}}$ is called simple if $\text {dim ker}(\lambda I - T) = 1$. Waelbroeck [42] considered the set $\rho ^{*}(T;X) (\subseteq \rho (T;X))$ consisting of all $\lambda \in {\mathbb {C}}$ for which there exists $\delta >0$ such that the open disk $B(\lambda , \delta ):= \{ z \in {\mathbb {C}}: |z-\lambda | < \delta \} \subseteq \rho (T;X)$ and $\{ R(\mu ;T): \mu \in B(\lambda , \delta ) \}$ is an equicontinuous subset in ${\mathcal {L}}(X)$. Then $\sigma ^{*}(T;X):= {\mathbb {C}} \setminus \rho ^{*}(T;X)$ is a closed set in ${\mathbb {C}}$, and satisfies $\overline{\sigma (T;X)} \subseteq \sigma ^{*}(T;X)$. Note that if X is a Banach space, then $\sigma (T;X)$ coincides with $\sigma ^{*}(T;X)$. For the classical spectral theory of compact operators in lcHs, we refer to [23, 27]. We know (see for example [8, Corollary 2.2]) that if X is a complete and barrelled lcHs and $T \in {\mathcal {L}}(X)$, then

$$\begin{aligned} \rho (T;X) = \rho (T'; X'_{\beta }), \qquad \sigma (T;X) = \sigma (T'; X'_{\beta }), \qquad \sigma ^{*}(T';X'_{\beta }) \subseteq \sigma ^{*}(T;X).\nonumber \\ \end{aligned}$$

(4.1)

The proof of this result is along the lines of that for (LB)-spaces given in [3, Lemma 5.2]:

Lemma 10

Let $X = \text {ind}_n X_n$ be an (LF)-space. Let $T \in {\mathcal {L}}(X)$ satisfy the following condition:

• (A) For each $n \in {\mathbb {N}}$, the restriction $T_n$ of T to $X_n$ maps $X_n$ into itself and $T_n \in {\mathcal {L}}(X_n)$.

Then, the following properties are satisfied:

• (i) $\displaystyle \sigma _{pt}(T;X) = \bigcup \nolimits _{n=1}^{\infty } \sigma _{pt}(T_n; X_n)$.

• (ii) $\displaystyle \sigma (T;X) \subseteq \bigcap \nolimits _{m \in {\mathbb {N}}} \Big ( \bigcup \nolimits _{n=m}^{\infty } \sigma (T_n; X_n)\Big )$.

• (iii) If $\displaystyle \bigcup \nolimits _{n=m}^{\infty } \sigma (T_n;X_n) \subseteq \overline{\sigma (T;X)}$ for some $m \in {\mathbb {N}}$, then $\sigma ^{*}(T; X) = \overline{\sigma (T;X)}$.

Observe that in the proof it is used the open mapping theorem (see for example [34]) which is valid in the setting of (LF)-spaces as they have a web and are ultrabornological.

Results regarding the spectra of (PLB)-spaces are stated and shown below (compare them with Lemma 10). To that aim, we need some preparation.

Lemma 11

Let X be a lcHs, $T \in {\mathcal {L}}(X)$ and $\lambda \in \rho ^{*}(T;X)$. If $\overline{B(\lambda , \varepsilon )} \subset \rho ^{*}(T;X)$ for some $\varepsilon >0$, then the set $\{ R(z,T): z \in \overline{B(\lambda , \varepsilon )} \}$ is equicontinuous.

Proof

Since $\overline{B(\lambda , \varepsilon )} \subset \rho ^{*}(T;X)$, for every $\mu \in \overline{B(\lambda , \varepsilon )}$, there exists $\varepsilon (\mu )>0$ such that $B(\mu , \varepsilon (\mu )) \subset \rho (T;X)$ and the set $\{ R(z, T): z \in B(\mu , \varepsilon (\mu )) \}$ is equicontinuous. Therefore,

$$\begin{aligned} \overline{B(\lambda , \varepsilon )} \subset \bigcup _{\mu \in \overline{B(\lambda , \varepsilon )}} B(\mu , \varepsilon (\mu )). \end{aligned}$$

Since $\overline{B(\lambda , \varepsilon )}$ is a compact subset of ${\mathbb {C}}$, there exist $\mu _1, \ldots , \mu _k \in \overline{B(\lambda , \varepsilon )}$ such that

$$\begin{aligned} \overline{B(\lambda , \varepsilon )} \subset \bigcup _{i=1}^k B(\mu _i, \varepsilon _i), \end{aligned}$$

(4.2)

with $\varepsilon _i:= \varepsilon (\mu _i)$ for $1 \le i \le k$.

Since the set $\{ R(z, T): z \in B(\mu _i, \varepsilon _i) \}$ is equicontinuous for every $1 \le i \le k$, fixed $p \in \Gamma _X$, for each $i=1,\ldots ,k$ there exist $q_i \in \Gamma _X$ and $M_i>0$ such that

$$\begin{aligned} p(R(z,T)x) \le M_iq_i(x), \qquad z \in B(\mu _i, \varepsilon _i), \ x \in X. \end{aligned}$$

Now, there exists $q \in \Gamma _X$ such that $\max \{ q_i(x): i = 1, \ldots , k \} \le q(x)$ for all $x \in X$. So, set $M:=\max \{ M_i: i = 1, \ldots , k \}$, it follows that

$$\begin{aligned} p(R(z,T)x) \le Mq(x), \qquad z \in \bigcup _{i=1}^k B(\mu _i, \varepsilon _i), \ x \in X. \end{aligned}$$

In view of (4.2), this shows that $\{ R(z, T): z \in \overline{B(\lambda , \varepsilon )} \}$ is equicontinuous. $\square$

We show the following result (cf. [2, Lemma 2.1] for Fréchet spaces):

Lemma 12

Let $X= \bigcap _{n=1}^{\infty } X_n$ be a barrelled (PLB)-space. Let $T \in {\mathcal {L}}(X)$ satisfy the following property:

• (A’) For every $n \in {\mathbb {N}}$ there exists $T_n \in {\mathcal {L}}(X_n)$ such that $T_n |_{X} = T$ and $T_n |_{X_{n+1}} = T_{n+1}$.

Then:

• (i) $\sigma (T;X) \subseteq \bigcup _{n=1}^{\infty } \sigma (T_n;X_n)$ and $\sigma _{pt}(T;X)\subseteq \bigcap _{n\in {\mathbb {N}}}\sigma _{pt}(T_n;X_n)$.

• (ii) For all $\lambda \in \bigcap _{n=1}^{\infty } \rho (T_n;X_n)$ the resolvent $R(\lambda , T)$ of T coincides with the restriction of $R(\lambda , T_n)$ of $T_n$ to X for each $n \in {\mathbb {N}}$.

• (iii) If $\bigcup _{n=1}^{\infty } \sigma ^{*}(T_n; X_n) \subseteq \overline{\sigma (T;X)}$, then $\sigma ^{*}(T;X) = \overline{\sigma (T;X)}$.

• (iv) If $\text {dim ker}(\lambda I-T_m)=1$ for each $\lambda \in \bigcap _{n\in {\mathbb {N}}}\sigma _{pt}(T_n;X_n)$ and for each $m\in {\mathbb {N}}$, then $\sigma _{pt}(T;X)=\bigcap _{n\in {\mathbb {N}}}\sigma _{pt}(T_n;X_n)$.

Proof

The proof of points (i) and (ii) is along the lines of [2, Lemma 2.1]. Indeed, we take $\lambda \in \bigcap _{n=1}^{\infty } \rho (T_n;X_n)$ and we show that $\lambda \in \rho (T;X)$. We see that $\lambda I - T: X\rightarrow X$ is injective: if $(\lambda I - T)x=0$ for some $x\in X$, then by (A’) we have $(\lambda I - T_1)x=0$ in $X_1$. Since $\lambda \in \rho (T_1;X_1)$, we have $x=0$. To show that $\lambda I - T$ is surjective, we fix $y \in X$. Since $\lambda I - T_n$ is surjective for each $n\in {\mathbb {N}}$, it follows that for every $n \in {\mathbb {N}}$ there exists $x_n \in X_n$ satisfying $(\lambda I - T_n)x_n = y$ in $X_n$ for every $n\in {\mathbb {N}}$. By condition (A’) we have that $T_n |_{X_{n+1}} = T_{n+1}$. Hence, $y=(\lambda I - T_n)x_n = (\lambda I - T_{n+1})x_{n+1} = (\lambda I - T_n)x_{n+1}$ in $X_{n+1} \subseteq X_n$. Since $\lambda \in \rho (T_n;X_n)$, we obtain $x_n = x_{n+1}$ for every $n\in {\mathbb {N}}$. So, $x_1 \in X$ and $y = (\lambda I - T)x_1$. Hence, there exists the inverse operator $(\lambda I-T)^{-1}:X\rightarrow X$. It remains to show that $(\lambda I-T)^{-1}\in {\mathcal {L}}(X)$, thereby implying that $R(\lambda ,T)=(\lambda I-T)^{-1}$. So, we observe that the proof above implies that the resolvent $R(\lambda ,T)$ of T coincides with the restriction of $R(\lambda , T_n)$ to X for each $n\in {\mathbb {N}}$. Since $R(\lambda , T_n)\in {\mathcal {L}}(X_n)$ for each $n\in {\mathbb {N}}$, by Lemma 8 it then follows that $R(\lambda ,T)\in {\mathcal {L}}(X)$. Accordingly, $\lambda \in \rho (T;X)$ as desired. Since $\lambda \in \bigcap _{n=1}^{\infty } \rho (T_n;X_n)$ is arbitrary, we conclude that $\bigcap _{n=1}^{\infty } \rho (T_n;X_n)\subseteq \rho (T;X)$ and hence, $\sigma (T;X) \subseteq \bigcup _{n=1}^{\infty } \sigma (T_n;X_n)$.

Finally, the proof of $\sigma _{pt}(T;X)\subseteq \bigcap _{n\in {\mathbb {N}}}\sigma _{pt}(T_n;X_n)$ is similar to that in [8, Lemma 2.5].

(iii) We have that $\overline{\sigma (T;X)} \subseteq \sigma ^{*}(T;X)$. If $\overline{\sigma (T;X)} = {\mathbb {C}}$, then there is nothing to prove. So, we suppose ${\mathbb {C}} {\setminus } \overline{\sigma (T;X)}\not =\emptyset$ and we take $\lambda \in {\mathbb {C}} {\setminus } \overline{\sigma (T;X)}$. Then, there exists $\varepsilon >0$ such that $\overline{B(\lambda , \varepsilon )} \cap \overline{\sigma (T;X)} = \emptyset$. By assumption, we have $\overline{B(\lambda , \varepsilon )} \cap \sigma ^{*}(T_n;X_n) = \emptyset$ for every $n \in {\mathbb {N}}$, that is, $\overline{B(\lambda , \varepsilon )} \subseteq \rho ^{*}(T_n;X_n)$ for every $n\in {\mathbb {N}}$. By Lemma 11, we have that $\{ R(\mu ,T_n): \mu \in \overline{B(\lambda , \varepsilon )} \}$ is equicontinuous in ${\mathcal {L}}_s(X_n)$ for every $n\in {\mathbb {N}}$. We claim that $\lambda \in \rho ^{*}(T;X)$. Since we know that $\overline{B(\lambda , \varepsilon )} \cap \sigma (T,X) = \emptyset$, we have $B(\lambda , \varepsilon ) \subseteq \overline{B(\lambda , \varepsilon )} \subseteq \rho (T;X)$. So, to show the claim, it is enough to see that $\{ R(\mu ,T)x: \mu \in B(\lambda , \varepsilon ) \}$ is bounded for every $x\in X$, as X is barrelled. By contradiction, we assume there exists $x \in X$ such that $\{ R(\mu , T)x: \mu \in \overline{B(\lambda , \varepsilon )} \}$ is an unbounded subset of X. Then, there is $n_0 \in {\mathbb {N}}$ such that the set $\{ R(\mu , T_{n_0})x: \mu \in \overline{B(\lambda ,\varepsilon )}\}$ is unbounded in $X_{n_0}$. This contradicts the fact that $\{ R(\mu , T_{n_0}): \mu \in \overline{B(\lambda , \varepsilon )} \}$ is equicontinuous in ${\mathcal {L}}_s(X_{n_0})$ by Lemma 11.

The proof of point (iv) follows as in [8, Lemma 2.5]. $\square$

4.1 Spectra of generalized Cesàro operators $C_t$ ($0 \le t < 1$)

The aim of this subsection is to study the spectra of the generalized Cesàro operators $C_t$, for $0 \le t < 1$, acting in the (LF)-spaces $L(p-)$, $C(p-)$, and $D(p-)$ ($1 < p \le \infty$) and in the (PLB)-spaces $L(p+)$, $C(p+)$, and $D(p+)$ ($1 \le p < \infty$). In order to do this, we first observe that

Proposition 13

Let $0\le t< 1$ and let X belong to $\{ L(p-), C(p-), D(p-); 1 < p \le \infty \}$ or to $\{ L(p+), C(p+), D(p+); 1 \le p < \infty \}$. Then, $C_t\in {\mathcal {L}}(X)$.

Proof

We first consider the case $X\in \{L(p-),C(p-),D(p-)\}$, with $1<p\le \infty$. So, we take a strictly increasing sequence $\{ p_k \}_{k\in {\mathbb {N}}}$ such that $1<p_k \uparrow p$ and set $X_k:=\ell (p_k+)$ if $X=L(p-)$ or $X_k:=\text {ces}(p_k+)$ ($X_k:=d(p_k+)$) if $X=C(p-)$ (if $X=D(p-)$), for any $k\in {\mathbb {N}}$. Then by [8, Proposition 4.4] we have $C_t\in {\mathcal {L}}(X_k)$ for every $k\in {\mathbb {N}}$. By Lemma 6 it necessarily follows that $C_t\in {\mathcal {L}}(X)$.

We now pass to consider the case that $X\in \{L(p+),C(p+),D(p+)\}$, with $1\le p<\infty$. So, we take a strictly decreasing sequence $\{ p_k \}_{k\in {\mathbb {N}}}$ such that $1<p_k \downarrow p$ and set $X_k:=\ell (p_k-)$ if $X=L(p+)$ or $X_k:=\text {ces}(p_k-)$ ($X_k:=d(p_k-)$) if $X=C(p+)$ (if $X=D(p+)$), for any $k\in {\mathbb {N}}$. Then by [8, Proposition 5.2] we have $C_t\in {\mathcal {L}}(X_k)$ for every $k\in {\mathbb {N}}$. By Lemma 8 it necessarily follows that $C_t\in {\mathcal {L}}(X)$. $\square$

We now turn our attention to the study of the spectra of $C_t$.

Lemma 14

Let $0 \le t < 1$ and let X belong to $\{ L(p-), C(p-), D(p-); 1 < p \le \infty \}$ or to $\{ L(p+), C(p+), D(p+); 1 \le p < \infty \}$. Then $0 \in \sigma (C_t;X)$.

Proof

Let $0 \le t < 1$ be fixed. By [8, Proposition 3.2] there exists the inverse operator $C^{-1}_t: \omega \rightarrow \omega$ and is given (see formula (3.5) in [8]) by

$$\begin{aligned} C^{-1}_t(x) = \big ( x_0, \big ((n+1)x_n - ntx_{n-1} \big )_{n\in {\mathbb {N}}}\big ), \qquad x=(x_0, x_1, \ldots ) \in \omega . \end{aligned}$$

(4.3)

Since $X\subseteq \omega$ with continuous inclusion, to show that $0\in \sigma (C_t;X)$ it suffices to establish that $C^{-1}_t(X)$ does not contain X.

We first consider the case that $X\in \{L(p-), C(p-), D(p-)\}$, with $1<p\le \infty$ fixed. So, we take a strictly increasing sequence $\{ p_k \}_{k\in {\mathbb {N}}}$ such that $1<p_k \uparrow p$ and strictly decreasing sequences $\{ p_{k,l} \}_{l\in {\mathbb {N}}}$ such that $p_k< p_{k,l} < p_{k+1}$, for $k,l\in {\mathbb {N}}$, and $p_{k,l} \downarrow p_k$. Now, we observe that the sequence

$$\begin{aligned} \varphi = (\varphi _n)_{n \in {\mathbb {N}}_0} = \Big ( \frac{1}{n+1} \Big )_{n \in {\mathbb {N}}_0} \end{aligned}$$

belongs to $L(p-)$. Indeed, for every $k,l\in {\mathbb {N}}$ we have

$$\begin{aligned} \left\Vert \varphi \right\Vert _{p_{k,l}} = \sum _{n=0}^{\infty } \Big (\frac{1}{n+1}\Big )^{p_{k,l}} < \infty , \end{aligned}$$

and hence, $\varphi \in \ell (p_k+) \subseteq L(p-)$. However, from (4.3) it follows that

$$\begin{aligned} C^{-1}_t(\varphi ) = \Big (\varphi _0, \Big (\frac{n+1}{n+1}-\frac{nt}{n}\Big )_{n \in {\mathbb {N}}}\Big ) = (1, 1-t, 1-t, \ldots ). \end{aligned}$$

(4.4)

Accordingly, for every $k,l\in {\mathbb {N}}$ we have

$$\begin{aligned} \sum _{n=0}^{\infty } |(C^{-1}_t(\varphi ))_n|^{p_{k,l}} = 1 + \sum _{n=1}^{\infty } (1-t)^{p_{k,l}} = +\infty , \end{aligned}$$

which implies that $C^{-1}_t(\varphi ) \notin L(p-)$.

Since $\varphi$ and $C^{-1}_t(\varphi )$ are decreasing sequences, we have that ${\hat{\varphi }} = \varphi$ and $\widehat{C^{-1}_t(\varphi )} = C^{-1}_t(\varphi )$. So, the same argument shows that $\varphi \in d(p_k+)\subseteq D(p-)$ (for every $k\in {\mathbb {N}}$), but, $C^{-1}_t(\varphi ) \not \in D(p-)$.

Since $D(p-)\subseteq C(p-)$ (see Proposition 7) and $\varphi \in D(p-)$, we also have that $\varphi \in C(p-)$. On the other hand, from (4.4) it follows that

$$\begin{aligned} C_1(C_t^{-1}(\varphi ))=\Big (1, \Big ( 1-\frac{n}{n+1}t \Big )_{n\in {\mathbb {N}}}\Big )=\Big (1, 1-\frac{1}{2}t, 1-\frac{2}{3}t,\ldots \Big ) \end{aligned}$$

and hence,

$$\begin{aligned} \sum _{n=0}^\infty |(C_1(C_t^{-1}(\varphi )))_n|^{p_{k,l}}=\sum _{n=0}^\infty \Big (1-\frac{n}{n+1}t\Big )^{p_{k,l}}=\infty , \end{aligned}$$

as $\left( 1-\frac{n}{n+1}t\right) ^{p_{k,l}}\rightarrow (1-t)^{p_{k,l}}\not =0$. This means that $C^{-1}_t(\varphi )\not \in C(p-)$.

We now pass to consider the case that $X\in \{L(p+), C(p+), D(p+)\}$, with $1\le p <\infty$ fixed. So, we take a strictly decreasing sequence $\{p_k\}_{k\in {\mathbb {N}}}$ such that $p_k \downarrow p$ and strictly increasing sequences $\{p_{k,l}\}_{l\in {\mathbb {N}}}$ such that $p_{k+1}< p_{k,l} < p_k$, for $k,l\in {\mathbb {N}}$, and $p_{k,l} \uparrow p_k$. Arguing as above, we obtain that $\varphi \in X$, but $C_t^{-1}(\varphi )\not \in X$. $\square$

We denote

$$\begin{aligned} \Lambda : = \Big \{ \frac{1}{n+1} \, \ n \in {\mathbb {N}}_0 \Big \}\quad \textrm{and}\quad \Lambda _0:=\Lambda \cup \{0\}. \end{aligned}$$

Theorem 15

Let $0 \le t < 1$ and let X belong to $\{ L(p-), C(p-), D(p-); 1 < p \le \infty \}$ or $\{ L(p+), C(p+), D(p+); 1 \le p < \infty \}$. Then

$$\begin{aligned} \sigma _{pt}(C_t; X) = \Lambda \qquad \text {and} \qquad \sigma (C_t; X) = \sigma ^{*}(C_t; X) = \Lambda _0. \end{aligned}$$

Moreover, every $\lambda \in \Lambda$ is a simple eigenvalue.

Proof

Let $X\in \{ L(p-), C(p-), D(p-)\}$ be fixed, with $1 < p \le \infty$. Then, for every $k\in {\mathbb {N}}$ we denote by $X_k$ the Fréchet space $\ell (p_k+)$ if $X=L(p-)$ or $\text {ces}(p_k+)$ ($d(p_k+)$) if $X=C(p-)$ (if $D(p-)$), where $\{p_k\}_{k\in {\mathbb {N}}}$ is any strictly increasing sequence satisfying $p_k\uparrow p$. By [8, Theorem 4.5] we have that

$$\begin{aligned} \sigma _{pt}(C_t; X_k) = \Lambda \qquad \text {and} \qquad \sigma (C_t; X_k) = \sigma ^{*}(C_t; X_k) = \Lambda _0. \end{aligned}$$

(4.5)

Since the restriction of $C_t$ to $X_k$ maps $X_k$ into itself for every $k\in {\mathbb {N}}$, we can apply Lemma 10(i)–(ii) to obtain that

$$\begin{aligned} \Lambda = \sigma _{pt}(C_t; X) \subseteq \sigma (C_t; X) \subseteq \Lambda _0. \end{aligned}$$

On the other hand, by Lemma 14 we have that $0\in \sigma (C_t;X)$. Therefore, it follows that $\sigma (C_t;X) = \Lambda _0$. Moreover, the assumption in Lemma 10(iii) is fulfilled, and hence

$$\begin{aligned} \sigma ^{*}(C_t; X) = \overline{\sigma (C_t; X)} = \Lambda _0. \end{aligned}$$

If $\lambda \in \sigma _{pt}(C_t;X)$, then $\{0\}\not =\ker (\lambda I-C_t)\subseteq \ker (\lambda I-C_t^\omega )$ (here, $C_t^\omega$ denotes the operator $C_t$ acting in $\omega$), as $X\subseteq \omega$. Accordingly, $0<\text {dim ker}(\lambda I-C_t)\le \text {dim ker}(\lambda I-C_t^\omega )=1$ (see [8, Lemma 3.4(i)]). It follows that $\text {dim ker}(\lambda I-C_t)=1$, i.e., $\lambda$ is a simple eigenvalue.

Now, we suppose $X\in \{L(p+), C(p+), D(p+)\}$, with $1\le p <\infty$. Then for every $k\in {\mathbb {N}}$ we denote by $X_k$ the (LB)-space $\ell (p_k-)$ if $X=L(p+)$ or $\text {ces}(p_k-)$ ($d(p_k-)$) if $X=C(p+)$ (if $X=D(p+)$), where $\{p_k\}_{k\in {\mathbb {N}}}$ is any strictly decreasing sequence satisfying $p_k\downarrow p$. By [8, Theorem 5.3] we have that (4.5) is valid also in this case. Moreover, by [8, Theorem 5.3] we also know that $\text {dim ker}(\frac{1}{n+1}I- C_t)=1$ in $X_k$, for each $k\in {\mathbb {N}}$ and $n \in {\mathbb {N}}_0$. Since $C_t$ maps $X_k$ into itself for every $k\in {\mathbb {N}}$, we can apply Lemma 12(iv) to conclude that

$$\begin{aligned} \sigma _{pt}(C_t;X)=\bigcap _{k\in {\mathbb {N}}}\sigma _{pt}(C_t;X_k)=\Lambda . \end{aligned}$$

By Lemma 12(i) we also obtain that

$$\begin{aligned} \sigma _{pt}(C_t; X) \subseteq \sigma (C_t; X) \subseteq \bigcup _{k=1}^{\infty } \sigma (C_t; X_k) = \Lambda _0. \end{aligned}$$

Since by Lemma 14 we have that $0\in \sigma (C_t;X)$, it then follows that $\sigma (C_t; X) = \Lambda _0$. Finally, by Lemma 12(ii)–(iii) we conclude that

$$\begin{aligned} \sigma ^{*}(C_t; X) = \overline{\sigma (C_t; X)} = \Lambda _0. \end{aligned}$$

The proof that every $\lambda \in \sigma _{pt}(C_t;X)$ is a simple eigenvalue follows as in the case of (LF)-spaces. $\square$

4.2 Spectra of the Cesàro operator

We are now concerned about the study of the operator $C_1$ and of its spectra. To do this, we first observe that

Proposition 16

Let X be one of the spaces in $\{ L(p-), C(p-), D(p-); 1 < p \le \infty \}$ or in $\{ L(p+), C(p+), D(p+); 1 \le p < \infty \}$. Then $C_1\in {\mathcal {L}}(X)$.

Proof

The proof is along the lines of the proof of Proposition 13, after having observed that the operator $C_1$ acts continuously in the Fréchet and (LB)-spaces considered in this paper, see [2, Sect. 2] for $\ell (p+)$, [4] for $\text {ces}(p+)$, [6, Proposition 5.3] for $\ell (p-)$ and $\text {ces}(p-)$, and [15, Proposition 4.9] for $d(p+)$ and $d(p-)$. $\square$

In the literature, the spectra of $C_1$ are analyzed, among others, when $C_1$ acts in the Fréchet and (LB)-spaces defined in (1.3) and (1.4). We state and refer to these results below. For $1 \le p < \infty$, we write

$$\begin{aligned} B(p/2, p/2) = \Big \{ z \in {\mathbb {C}} \, \ \Big | z - \frac{p}{2} \Big | < \frac{p}{2} \Big \}. \end{aligned}$$

Lemma 17

Let X belong to $\{ \ell (p+), \text {ces}(p+), d(p+); 1 \le p < \infty \}$. If $1< p < \infty$, then

$$\begin{aligned} \sigma _{pt}(C_1; X)&= \emptyset \qquad \text {and} \qquad B(p'/2, p'/2) \subseteq \sigma _{pt}(C'_1, X'_{\beta }); \\ \sigma (C_1; X)&= B(p'/2, p'/2) \cup \{0\}; \\ \sigma ^{*}(C_1; X)&= \overline{\sigma (C_1; X)} = \overline{B(p'/2, p'/2)}, \end{aligned}$$

where $1<p'<\infty$ satisfies $1/p + 1/p' = 1$. On the other hand, if $p=1$, then

$$\begin{aligned} \sigma _{pt}(C_1; X)&= \emptyset \qquad \text {and} \qquad \{ z \in {\mathbb {C}} : Rez> 0 \} \subseteq \sigma _{pt}(C'_1; X'_{\beta }); \\ \sigma (C_1; X)&= \{ z \in {\mathbb {C}} : Rez>0 \} \cup \{0\}; \\ \sigma ^{*}(C_1; X)&= \overline{\sigma (C_1; X)} = \{ z \in {\mathbb {C}} : Rez\ge 0 \}. \end{aligned}$$

Moreover, every $0 \ne \lambda \in \sigma (C_1; X)$ is a simple eigenvalue for $C'_1$ acting in $X'_\beta$.

For the proof of the results in Lemma 17, we refer the reader to [2, Theorem 2.2] for $\ell (p+)$ ($1<p<\infty$), to [2, Theorem 2.4] for $\ell (1+)$, to [7, Theorem 2.3 and Proposition 2.4] for $\text {ces}(p+)$ ($1 \le p < \infty$), and to [16, Theorem 3.2 and Proposition 3.3] for $d(p+)$ ($1 \le p < \infty$).

Lemma 18

Let X belong to $\{ \ell (p-), \text {ces}(p-), d(p-); 1 < p \le \infty \}$. Then

$$\begin{aligned}&\sigma _{pt}(C_1; X) = \emptyset \qquad \text {and} \qquad B(p'/2, p'/2) \subseteq \sigma _{pt}(C'_1; X'_{\beta }); \\&B(p'/2, p'/2) \cup \{0\} \subseteq \sigma (C_1; X) \subseteq \overline{B(p'/2, p'/2)}; \\&\sigma ^{*}(C_1; X) = \overline{\sigma (C_1; X)} = \overline{B(p'/2, p'/2)}. \end{aligned}$$

where $1\le p' <\infty$ satisfies $1/p + 1/p' = 1$.

The proof of Lemma 18 for $\ell (p-)$ is given in [8, Proposition 5.5], for $\text {ces}(p-)$ is given in [6, Propositions 3.1, 3.2, 3.3], and for $d(p-)$ in [16, Theorem 3.6].

Let us begin the study of the spectra by considering $X \in \{ L(p-), C(p-), D(p-)\}$, with $1 < p \le \infty$.

Lemma 19

Let X belong to $\{ L(p-), C(p-), D(p-); 1 < p \le \infty \}$. Then $0 \in \sigma (C_1;X)$.

Proof

The formula in (4.3) is valid also for $t=1$, i.e., the inverse operator $C_1^{-1}:\omega \rightarrow \omega$ exists in ${\mathcal {L}}(\omega )$ and it is given by

$$\begin{aligned} C^{-1}_1(x) = \big ( x_0, \big ( (n+1)x_n - nx_{n-1} \big )_{n \in {\mathbb {N}}} \big ), \qquad x=(x_n)_n \in \omega . \end{aligned}$$

We consider the following sequence as in the proof of [6, Proposition 3.2]:

$$\begin{aligned} x = (x_n)_n = \Big ( \frac{1-(-1)^{n+1}}{2(n+1)} \Big )_{n \in {\mathbb {N}}_0} = (1, 0, 1/3, 0, 1/5, \ldots ). \end{aligned}$$

(4.6)

Therefore,

$$\begin{aligned} {\hat{x}} = (1,1/3,1/3,1/5,1/5,\ldots ). \end{aligned}$$

(4.7)

Now, let $X\in \{L(p-),C(p-),D(p-)\}$ be fixed, with $1<p\le \infty$, and let $\{p_k\}_{k\in {\mathbb {N}}}$ be a strictly increasing sequence satisfying $1<p_k\uparrow p$. Then for every $k\in {\mathbb {N}}$ we denote by $X_k$ the k-th step of the inductive spectrum defining X as done in the previous subsection (i.e., $X_k$ is one of the Fréchet spaces $\ell (p_k+)$, $\text {ces}(p_k+)$, $d(p_k+)$). So, for every $k\in {\mathbb {N}}$ we have the sequence x defined in (4.6) satisfies $x\in d(p_k+)\subseteq \ell (p_k+) \subseteq \text {ces}(p_k+)$ (see (4.7)) and hence, $x\in X_k\subseteq X$. However, the n-th entry, for $n \in {\mathbb {N}}$, of $C^{-1}_1(x)$ is given by

$$\begin{aligned} (n+1)\frac{(1-(-1)^{n+1})}{2(n+1)}-n\frac{(1-(-1)^n)}{2n} = \frac{(-1)^n+(-1)^n}{2} = (-1)^n. \end{aligned}$$

Since $x_0=1$, we get that $C^{-1}_1(|x|) = (|(-1)^n|)_{n \in {\mathbb {N}}_0}$, which does not belong to $\text {ces}(p_k+)$ for all $k\in {\mathbb {N}}$, as $C_1(|C^{-1}_1(x)|)=(1,1,1,1,1,\ldots )$. Accordingly, $C^{-1}_1(|x|)$ does not belong to either $d(p_k+)$ or to $\ell (p_k+)$ for all $k\in {\mathbb {N}}$. Therefore, $C^{-1}_1(|x|)\not \in X$. Thus, we obtain $0 \in \sigma (C_1; X)$ as we wanted. $\square$

In the following result, we see that the behaviour of the spectra of $C_1$ in the (LF)-spaces considered in this paper is similar to that in the (LB)-spaces in (1.4) (cf. Lemma 18):

Theorem 20

Let X belong to $\{ L(p-), C(p-), D(p-); 1<p\le \infty \}$. Then

$$\begin{aligned}&\sigma _{pt}(C_1; X) = \emptyset \qquad \text {and} \qquad B(p'/2, p'/2) \subseteq \sigma _{pt}(C'_1; X'_{\beta }); \\&B(p'/2, p'/2) \cup \{0\} \subseteq \sigma (C_1; X)\subseteq \overline{B(p'/2,p'/2)}; \\&\sigma ^{*}(C_1; X) = \overline{\sigma (C_1; X)} = \overline{B(p'/2, p'/2)}, \end{aligned}$$

where $1 \le p' < \infty$ satisfies $1/p+1/p'=1$.

Proof

Let $X\in \{L(p-),C(p-),D(p-)\}$ be fixed, with $1<p\le \infty$. So, let $\{p_k\}_{k\in {\mathbb {N}}}$ be a strictly increasing sequence satisying $1<p_k\uparrow p$. Then for every $k\in {\mathbb {N}}$ we denote by $X_k$ the k-th step of the inductive spectrum defining X as done in the previous subsection (i.e., $X_k$ is one of the Fréchet spaces $\ell (p_k+)$, $\text {ces}(p_k+)$, $d(p_k+)$). We observe that the conjugate exponents $p'_k$ form a strictly decreasing sequence such that $p'_k\downarrow p'$, where $1\le p'<\infty$ satisfies $1/p+1/p'=1$. We point out that also in the case $p=\infty$, we have $1<p_k<\infty$ for every $k\in {\mathbb {N}}$. Since the assumptions in Lemma 10 are satisfied, we can apply Lemma 10(i)–(ii) combined with Lemma 17 to deduce that $\sigma _{pt}(C_1; X) = \emptyset$ and

$$\begin{aligned} \sigma (C_1; X) \subseteq \bigcap _{m\in {\mathbb {N}}}\Big ( \bigcup _{k=m}^{\infty } \sigma (C_1; X_k)\Big )= & {} \bigcap _{m \in {\mathbb {N}}} \big ( B(p'_m/2, p'_m/2) \cup \{0\} \big ) \nonumber \\= & {} \overline{B(p'/2, p'/2)}. \end{aligned}$$

(4.8)

Accordingly,

$$\begin{aligned} \overline{\sigma (C_1; X)}\subseteq \overline{B(p'/2, p'/2)}. \end{aligned}$$

We now show that $B(p'/2,p'/2)\subseteq \sigma _{pt}(C'_1;X'_\beta )$. To this end, we recall that by Corollaries 3 and 5 we have that the strong dual $X'_\beta$ of X is reflexive (hence barrelled) and it is given by $X'_\beta =\bigcap _{k\in {\mathbb {N}}}(X_k)'_\beta$, where each $(X_k)'_\beta$ is one of the (LB)-spaces $\ell (p'_k-)$, $\text {ces}(p'_k-)$ or $d(p'_k-)$. On the other hand, by Lemma 17 we have that $B(p_k'/2,p'_k/2)\subseteq \sigma _{pt}(C'_1;(X_k)'_\beta )$ for every $k\in {\mathbb {N}}$. Since the assumptions in Lemma 12 are clearly satisfied with $T=C'_1$ and every element in $B(p'/2, p'/2)$ is a simple eigenvalue for $C'_1$, we can apply Lemma 12(iv) to deduce that

$$\begin{aligned} B(p'/2,p'/2) \subseteq \bigcap _{k \in {\mathbb {N}}} B(p'_k/2, p'_k/2) \subseteq \bigcap _{k\in {\mathbb {N}}}\sigma _{pt}(C'_1; (X_k)'_\beta ) = \sigma _{pt}(C'_1;X'_\beta ). \end{aligned}$$

So, it follows via (4.1) that

$$\begin{aligned} B(p'/2,p'/2) \subseteq \sigma _{pt}(C'_1;X'_\beta ) \subseteq \sigma (C'_1;X'_\beta ) = \sigma (C_1;X). \end{aligned}$$

Since $0\in \sigma (C_1;X)$ by Lemma 19, it follows by (4.8) that

$$\begin{aligned} B(p'/2,p'/2) \cup \{0\} \subseteq \sigma (C_1;X) \subseteq \overline{B(p'/2,p'/2)}. \end{aligned}$$

Since $\sigma (C_1;X) \subseteq \overline{B(p'/2,p'/2)}$ and $\overline{\sigma (C_1;X)}\subseteq \sigma ^{*}(C_1;X)$, we can argue as the proof of [6, Proposition 3.3] to conclude that

$$\begin{aligned} \sigma ^{*}(C_1; X) = \overline{\sigma (C_1; X)} = \overline{B(p'/2, p'/2)}. \end{aligned}$$

$\square$

We now pass to study the spectra of $C_1$ acting in the (PLB)-spaces $L(p+)$, $C(p+)$, and $D(p+)$, for $1\le p <\infty$. We first show that the analogous of Lemma 19 holds also in this case. The argument is similar.

Lemma 21

Let X belong to $\{ L(p+), C(p+), D(p+); 1 \le p < \infty \}$. Then $0 \in \sigma (C_1;X)$.

Proof

Let $X \in \{L(p+), C(p+), D(p+)\}$ be fixed, with $1\le p < \infty$ and let $\{p_k\}_{k\in {\mathbb {N}}}$ be a strictly decreasing sequence satisfying $p<p_k\downarrow p$. Then for every $k\in {\mathbb {N}}$ we denote by $X_k$ the k-th step of the projective spectrum defining X as done in the previous subsection (i.e., $X_k$ is one of the (LB)-spaces $\ell (p_k-)$, $\text {ces}(p_k-)$, $d(p_k-)$ with $p_k>p\ge 1$).

We now observe that the sequence x defined in (4.6) satisfies $x\in d(p_k-)\subseteq \ell (p_k-)\subseteq \text {ces}(p_k-)$ (see (4.7)) for all $k\in {\mathbb {N}}$, as each $p_k>1$, and hence, $x\in X_k$ for all $k\in {\mathbb {N}}$. Accordingly, $x\in X$. But, $C_1^{-1}(|x|)=(1)_{n\in {\mathbb {N}}_0}$ (see the proof of Lemma 19) and so, $C_1^{-1}(|x|)$ does not belong to $\text {ces}(p_k-)$ for all $k\in {\mathbb {N}}$, as $C_1(|C_1^{-1}(x)|)=(1)_{n\in {\mathbb {N}}}$. This implies that $C_1^{-1}(x)\not \in X$. Therefore, we deduce that $0\in \sigma (C_1,X)$. $\square$

We are ready to study the spectra of $C_1$ in the (PLB) sequence spaces considered.

Theorem 22

Let X belong to $\{ L(p+), C(p+), D(p+); 1\le p <\infty \}$. If $1<p<\infty$, then

$$\begin{aligned}&\sigma _{pt}(C_1; X) = \emptyset \qquad \text {and} \qquad B(p'/2, p'/2) \subseteq \sigma _{pt}(C'_1; X'_{\beta }); \\&B(p'/2, p'/2) \cup \{0\} \subseteq \sigma (C_1; X) \subseteq \overline{B(p'/2, p'/2)}; \\&\sigma ^{*}(C_1; X) = \overline{\sigma (C_1; X)} = \overline{B(p'/2, p'/2)}, \end{aligned}$$

where $1<p'<\infty$ satisfies $1/p+1/p'=1$. On the other hand, if $p=1$, then

$$\begin{aligned}&\sigma _{pt}(C_1; X) = \emptyset \qquad \text {and} \qquad \{z\in {\mathbb {C}}:\ Rez>0\} \subseteq \sigma _{pt}(C'_1; X'_{\beta }); \\&\{z\in {\mathbb {C}}:\ Rez>0\} \cup \{0\} \subseteq \sigma (C_1; X) \subseteq \{z\in {\mathbb {C}}:\ Rez\ge 0\}; \\&\sigma ^{*}(C_1; X) = \overline{\sigma (C_1; X)} = \{z\in {\mathbb {C}}:\ Rez\ge 0\}. \end{aligned}$$

Proof

Let $X\in \{L(p+),C(p+),D(p+)\}$ be fixed, with $1 \le p <\infty$ and let $\{p_k\}_{k\in {\mathbb {N}}}$ be a strictly decreasing sequence satisfying $p_k\downarrow p$, with $p<p_k$ for every $k\in {\mathbb {N}}$. Then for every $k\in {\mathbb {N}}$ we denote by $X_k$ the k-th step of the projective spectrum defining X as done in the previous subsection (i.e., $X_k$ is one of the (LB)-spaces $\ell (p_k-)$, $\text {ces}(p_k-)$, $d(p_k-)$ with $p_k>p\ge 1$). We observe that the conjugate exponent $p'_k$ forms a strictly increasing sequence such that $p'_k\uparrow p'$ with $1<p'_k<p'$, where $1<p'\le \infty$.

Fix $p\ne 1$. Since the assumptions in Lemma 12 are clearly satisfied, we can apply Lemma 12(i) combined with Lemma 18 to deduce that

$$\begin{aligned} \sigma _{pt}(C_1;X) = \bigcap _{k\in {\mathbb {N}}}\sigma _{pt}(C_1;X_k) = \emptyset \end{aligned}$$

and that

$$\begin{aligned} \sigma (C_1;X) \subseteq \bigcup _{k\in {\mathbb {N}}}\sigma (C_1;X_k) \subseteq \bigcup _{k\in {\mathbb {N}}}\overline{B(p'_k/2,p'_k/2)} \subseteq \overline{B(p'/2,p'/2)}. \end{aligned}$$

We now show that $B(p'/2,p'/2)\subseteq \sigma _{pt}(C'_1;X'_\beta )$. To this end, we recall that by Proposition 4 and Corollary 5, the (PLB)-space X is reflexive and its strong dual $X'_\beta$ is given by $X'_\beta =\bigcup _{k\in {\mathbb {N}}}(X_k)_\beta '$, where each $(X_k)'_\beta$ is one of the Fréchet spaces $\ell (p'_k+)$, $\text {ces}(p'_k+)$ or $d(p'_k+)$. On the other hand, by Lemma 18 we have that $B(p'_k/2,p'_k/2)\subseteq \sigma _{pt}(C'_1; (X_k)'_\beta )$ for every $k\in {\mathbb {N}}$. Since the assumption in Lemma 10 is clearly satisfied with $T=C_1'$, we can apply Lemma 10(i) to deduce that

$$\begin{aligned} B(p'/2,p'/2) = \bigcup _{k\in {\mathbb {N}}} B(p'_k/2,p'_k/2) \subseteq \bigcup _{k\in {\mathbb {N}}}\sigma _{pt}(C_1'; (X_k)'_\beta ) = \sigma _{pt}(C_1';X'_\beta ). \end{aligned}$$

So, it follows by (4.1) that

$$\begin{aligned} B(p'/2,p'/2) \subseteq \sigma _{pt}(C_1';X'_\beta ) \subseteq \sigma (C'_1;X'_\beta ) = \sigma (C_1;X). \end{aligned}$$

Since $0\in \sigma (C_1;X)$ by Lemma 21, we conclude that

$$\begin{aligned} B(p'/2,p'/2) \cup \{0\} \subseteq \sigma (C_1;X) \subseteq \overline{B(p'/2,p'/2)} \end{aligned}$$

and hence,

$$\begin{aligned} \overline{\sigma (C_1;X)}= \overline{B(p'/2,p'/2)}. \end{aligned}$$

Since by Lemma 18 we have $\sigma ^{*}(C_1; X_k) = \overline{B(p'_k/2,p'_k/2)}$ for every $k\in {\mathbb {N}}$, it follows that

$$\begin{aligned} \bigcup _{k\in {\mathbb {N}}}\sigma ^{*}(C_1;X_k) = \bigcup _{k\in {\mathbb {N}}}\overline{B(p'_k/2,p'_k/2)}\subseteq \overline{B(p'/2,p'/2)} = \overline{\sigma (C_1;X)}. \end{aligned}$$

We can then apply Lemma 12(iii) to conclude that $\sigma ^{*}(C_1;X) = \overline{\sigma (C_1;X)} = \overline{B(p'/2,p'/2)}$.

Now, consider $p=1$. In such a case, the result follows by arguing as above and observing that

$$\begin{aligned} \bigcup _{k\in {\mathbb {N}}} B(p'_k/2,p'_k/2) = \{z\in {\mathbb {C}}:\ Rez>0\}, \end{aligned}$$

as $p'_k\rightarrow \infty$. $\square$

5 Dynamics of generalized Cesàro operators in (LF) and (PLB)-spaces

Let X be a lcHs. For $T \in {\mathcal {L}}(X)$ and $n \in {\mathbb {N}}$, we write $T^n = T \circ \cdots \circ T$. The Cesàro means of T are denoted by

$$\begin{aligned} T_{[n]} = \frac{1}{n} \sum _{m=1}^n T^m. \end{aligned}$$

We say that T is

• (1) power bounded if $\{ T^n \}_{n \in {\mathbb {N}}}$ is equicontinuous in ${\mathcal {L}}(X)$;

• (2) (uniformly) mean ergodic if $\{ T_{[n]} \}_{n \in {\mathbb {N}}}$ converges in ${\mathcal {L}}_s(X)$ (in ${\mathcal {L}}_b(X)$).

For a separable lcHs X, we say that T is

• (3) hypercyclic if there exists $x\in X$ whose orbit $\{ T^n x \, \ n \in {\mathbb {N}}_0\}$ is dense in X;

• (4) supercyclic if there exists $x\in X$ such that the projective orbit $\{ \lambda T^n x \, \ \lambda \in {\mathbb {C}}, n \in {\mathbb {N}}_0\}$ is dense in X.

We refer the reader to [11, 26] for general textbooks.

5.1 Dynamics of generalized Cesàro operators $C_t$ ($0 \le t < 1$)

Proposition 23

Let $0 \le t < 1$ and let X belong to $\{ L(p-), C(p-), D(p-); 1 < p \le \infty \}$ or to $\{ L(p+), C(p+), D(p+); 1 \le p < \infty \}$. Then the generalized Cesàro operator $C_t$ is not supercyclic in X.

Proof

Since the operator $C_t$ is not supercyclic in $\omega$ by [8, Theorem 6.1(iii)], it follows that $C_t$ cannot be supercyclic in X, as $C_t$ continuously maps X into itself and X is dense in $\omega$. $\square$

To study the power boundedness and the mean ergodicity of $C_t$ acting in the (LF)-spaces (in the (PLB)-spaces) considered in this paper, we first establish some results on continuous linear operators to compare their ergodic properties with the ones of their inductive spectrum (projective spectrum).

We first consider the case of operators acting in (LF)-spaces.

Theorem 24

Let $X = \text {ind}_k X_k=\bigcup _{k\in {\mathbb {N}}}X_k$ be an (LF)-space such that the inclusion $X_k \subseteq X_{k+1}$ is continuous, for $k\in {\mathbb {N}}$, and let $T \in {\mathcal {L}}(X)$ satisfy assumption (A) of Lemma 10. Then the following properties are satisfied.

• (i) If $T_k:=T|_{X_k}$ is power bounded in $X_k$ for every $k\in {\mathbb {N}}$, then T is power bounded in X.

• (ii) If $T_k$ is mean ergodic in $X_k$ for every $k\in {\mathbb {N}}$, then T is mean ergodic in X.

• (iii) If $T_k$ is uniformly mean ergodic in $X_k$ for every $k\in {\mathbb {N}}$ and X is regular, then T is uniformly mean ergodic in X.

Proof

• (i) Let $x \in X$ be fixed. Then there exists $k \in {\mathbb {N}}$ so that $x \in X_k$. Since $T(X_k)=T_k(X_k) \subseteq X_k$, we have that $T^nx \in X_k$ for every $n \in {\mathbb {N}}$. But, $T=T_k$ is power bounded in $X_k$. So, we have that $\{ T^nx: n \in \mathbb {N} \}$ is bounded in $X_k$, and hence, it is in X. Since $x \in X$ is arbitrary, and X is barrelled, we conclude that $\{ T^n \}_{n \in {\mathbb {N}}}$ is equicontinuous, i.e., T is power bounded in X.

• (ii) Let $x \in X$ be fixed. Then there exists $k \in {\mathbb {N}}$ such that $x \in X_k$. Since $T=T_k:X_k\rightarrow X_k$ is mean ergodic, we have that $\{ T_{[n]}x \}$ is convergent in $X_k$ and hence, $\{ T_{[n]}x \}$ is also convergent in X, as the inclusion $X_k \subseteq X$ is continuous. Since $x \in X$ is arbitrary, we can conclude that T is mean ergodic in X.

• (iii) We assume that X is regular and $T_k=T|_{X_k}$ is uniformly mean ergodic in each $X_k$. By (ii), we have that $T_{[n]}$ converges to some $P \in {\mathcal {L}}(X)$ in ${\mathcal {L}}_s(X)$. We fix $B \in {\mathcal {B}}(X)$. Then, there is $k \in {\mathbb {N}}$ so that $B \in {\mathcal {B}}(X_k)$. By assumption on $T \in {\mathcal {L}}(X_k)$, we have that for every $s \in \Gamma _k=\Gamma _{X_k}$,

  $$\begin{aligned} \sup _{x \in B} s(T_{[n]}x - Px) \rightarrow 0 \qquad \text {as} \ n\rightarrow \infty . \end{aligned}$$

  Now, we take $r \in \Gamma _X$. Since $X_k \subseteq X$ with continuous inclusion, there exists $C>0$ and $s \in \Gamma _k = \Gamma _{X_k}$ so that

  $$\begin{aligned} r(x) \le C s(x), \qquad x \in X_k. \end{aligned}$$

  Therefore,

  $$\begin{aligned} \sup _{x \in B} r(T_{[n]}x - Px) \le C \sup _{x \in B} s(T_{[n]}-Px), \end{aligned}$$

  from which it follows

  $$\begin{aligned} \sup _{x \in B} r(T_{[n]}x - Px) \rightarrow 0, \qquad \text {as} \ n \rightarrow \infty . \end{aligned}$$

  This completes the proof of point (iii). $\square$

Corollary 25

Let $0 \le t < 1$ and let X belong to $\{ L(p-), C(p-), D(p-); 1 < p \le \infty \}$. Then the generalized Cesàro operator $C_t$ is power bounded and uniformly mean ergodic in X.

Proof

The result follows by Theorem 24 and [8, Theorem 6.6]. $\square$

We now pass to the case of operators acting in (PLB)-spaces.

Theorem 26

Let $X = \text {proj}_k X_k=\bigcap _{k\in {\mathbb {N}}} X_k$ be a (PLB)-space and let $T \in {\mathcal {L}}(X)$ satisfy the assumption (A’) in Lemma 12, i.e., for every $k\in {\mathbb {N}}$ there exists $T_k\in {\mathcal {L}}(X_k)$ such that $T_k|_X = T$ and $T_k|_{X_{k+1}} = T_{k+1}$. If $T_k$ is power bounded ((uniformly) mean ergodic) in $X_k$ for every $k\in {\mathbb {N}}$, then T is power bounded ((uniformly) mean ergodic) in X.

Proof

Let us first prove the power boundedness. We write $\Gamma _k$ to denote a fundamental system of seminorms in $X_k$, for $k \in {\mathbb {N}}$, and we set

$$\begin{aligned} \Gamma := \Big \{ \max _{i=1,\ldots ,k} r_i: k \in \mathbb {N}, \ r_i \in \Gamma _i, \ i=1,\ldots ,k \Big \}. \end{aligned}$$

Then $\Gamma$ is a fundamental system of seminorms in X.

Fixed $r \in \Gamma$, there exist $k \in {\mathbb {N}}$ and $r_i \in \Gamma _i$ for $i=1,\ldots ,k$ such that $r = \max _{i=1,\ldots ,k} r_i$. Since $T_i$ is power bounded in $X_i$, for $i=1,\ldots ,k$, we have that for every $i=1,\ldots ,k$, there exist $C_i>0$ and $s_i \in \Gamma _i$ such that

$$\begin{aligned} r_i(T^n_i x) \le C_i s_i(x), \qquad x \in X_i, \ n \in {\mathbb {N}}. \end{aligned}$$

Thus, for $C:=\max _{i=1,\ldots ,k} C_i$, we have

$$\begin{aligned} r(T^nx) = \max _{i=1,\ldots ,k} r_i(T^n_ix) \le \max _{i=1,\ldots ,k} C_i s_i(x) \le C \max _{i=1,\ldots ,k} s_i(x) \end{aligned}$$

for every $x \in X$ and $n \in {\mathbb {N}}$, as $T_i|_{X}=T$ for $i=1,\ldots ,k$. Therefore, T is power bounded in X.

We now pass to consider the mean ergodicity. So, we fix $x \in X$. Then, $x \in X_k$ for every $k\in {\mathbb {N}}$. Since $T_k:X_k \rightarrow X_k$ is mean ergodic for every $k\in {\mathbb {N}}$, $(T_k)_{[n]}x$ converges to some $y_k$ in $X_k$ for every $k\in {\mathbb {N}}$. For $k=2$, we have that $(T_2)_{[n]}x$ converges to $y_2$ in $X_2$, and also in $X_1$, as $X_2\subseteq X_1$ with continuous inclusion. But, $(T_2)_{[n]}x=(T_1)_{[n]}x$ (as $x\in X\subseteq X_k$ for every k) converges to $y_1$ in $X_1$. Thus $y_2=y_1$. Proceeding inductively, we can see that all the $y_k$ coincide, and denoting it by y, we have that $y\in X$ and $T_{[n]}x$ converges to y in X.

Finally, we consider the uniformly mean ergodicity. So, we fix $B \in {\mathcal {B}}(X)$. Then $B \in {\mathcal {B}}(X_k)$ for every $k\in {\mathbb {N}}$. Since $T_k:X_k\rightarrow X_k$ is uniformly mean ergodic for every $k\in {\mathbb {N}}$, there exists $P_k\in {\mathcal {L}}(X_k)$ such that for every $r_k \in \Gamma _k$ we have (for every $k\in {\mathbb {N}}$ we have $T=T_k$ on B as $B\subseteq X$)

$$\begin{aligned} \sup _{x \in B} r_k(T_{[n]}x - P_kx)=\sup _{x \in B} r_k((T_k)_{[n]}x - P_kx) \rightarrow 0. \end{aligned}$$

This yields that $T:X \rightarrow X$ is uniformly mean ergodic in X and $P_k|_X=P_{k+1}|_X$ for every $k\in {\mathbb {N}}$. In particular, the operator $P:X\rightarrow X$ defined by $Px:=P_1x$ for $x\in X$ belongs to ${\mathcal {L}}(X)$ and $T_{[n]}\rightarrow P$ in ${\mathcal {L}}_b(X)$ as $n\rightarrow \infty$. $\square$

Corollary 27

Let $0 \le t < 1$ and let X belong to $\{ L(p+), C(p+), D(p+); 1 \le p < \infty \}$. Then the generalized Cesàro operator $C_t$ is power bounded and uniformly mean ergodic in X.

Proof

The result follows as an immediate application of Theorem 26 and [8, Theorem 6.6]. $\square$

5.2 Dynamics of the Cesàro operator

It is known that the Cesàro operator $C_1$ is neither power bounded nor mean ergodic in $\ell _p$ ($1<p<\infty$) (see [1, Proposition 4.2]), and it cannot be supercyclic since $\sigma _{pt}(C'_1; \ell _{p'})$ contains too many elements (see [11, Proposition 1.26]). Furthermore, the same characteristics hold for $\text {ces}(p)$ ($1<p<\infty$) (see [5, Proposition 3.7(ii)]) and for d(p) ($1<p<\infty$) (see [14, Propositions 3.10 and 3.11]).

For the Fréchet spaces and (LB)-spaces defined in (1.3) and (1.4), we have that $C_1$ is not mean ergodic nor power bounded nor supercyclic in $\ell (p+)$ ($1 \le p < \infty$) [2, Theorems 2.3 and 2.5], in $\text {ces}(p+)$ ($1 \le p < \infty$) [7, Proposition 5], in $d(p+)$ ($1 \le p < \infty$) [16, Proposition 3.5], in $\ell (p-)$ ($1 < p \le \infty$) [8, Proposition 6.10], in $\text {ces}(p-)$ ($1<p\le \infty$) [6, Propositions 3.4, 3.5], and in $d(p-)$ ($1<p \le \infty$) [16, Proposition 3.8].

On the other hand, the dynamics of $C_1$ in $\omega$ are the same as the ones for $C_t$ ($0 \le t < 1$) in $\omega$. Indeed, by [7, Theorem 6.1] and [2, Proposition 4.3], we have that $C_1$ is power bounded, mean ergodic, and not supercyclic in $\omega$.

The proof of the following result is similar to the ones from the references above.

Proposition 28

Let X belong to $\{ L(p-), C(p-), D(p-), L(q+), C(q+), D(q+); \ 1< p \le \infty , \ 1 \le q < \infty \}$. Then $C_1$ is neither power bounded nor mean ergodic nor supercyclic in X.

Proof

By Theorems 20 and 22 we have that $B(p'/2, p'/2)$, for $1<p\le \infty$ ($B(q'/2, q'/2)$, for $1<q<\infty$ and $\{ z \in {\mathbb {C}} \! \, \! \ Rez > 0 \}$ for $q=1$) are included in $\sigma _{pt}(C'_1; X'_{\beta })$. So, by [11, Proposition 1.26] we conclude that $C_1$ cannot be supercyclic in X. Furthermore, for $p,q<\infty$,

$$\begin{aligned} \sigma _{pt}(C'_1; X'_{\beta }) \cap \{ \lambda \in \mathbb {C}: \ |\lambda | > 1 \} \ne \emptyset . \end{aligned}$$

Thus, by [6, Lemma 3.2], $C_1$ is neither power bounded nor mean ergodic. The proof for $p=\infty$ is the same as in [6, Proposition 3.4]. $\square$