Abstract
An investigation is made of the generalized Cesàro operators \(C_t\), for \(t\in [0,1]\), when they act on the space \(H({{\mathbb {D}}})\) of holomorphic functions on the open unit disc \({{\mathbb {D}}}\), on the Banach space \(H^\infty \) of bounded analytic functions and on the weighted Banach spaces \(H_v^\infty \) and \(H_v^0\) with their sup-norms. Of particular interest are the continuity, compactness, spectrum and point spectrum of \(C_t\) as well as their linear dynamics and mean ergodicity.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction and preliminaries
The (discrete) generalized Cesàro operators \(C_t\), for \(t\in [0,1]\), were first investigated by Rhaly [25, 26]. The action of \(C_t\) from the sequence space \(\omega := {{\mathbb {C}}}^{{{\mathbb {N}}}_0}\) into itself, with \({{\mathbb {N}}}_0:=\{0,1,2,\ldots \}\), is given by
For \(t=0\) and with \(\varphi :=(\frac{1}{n+1})_{n\in {{\mathbb {N}}}_0}\) note that \(C_0\) is the diagonal operator
and, for \(t=1\), that \(C_1\) is the classical Cesàro averaging operator
The behaviour of \(C_t\) on various sequence spaces has been investigated by many authors. We refer the reader to [25,26,27], to the recent papers [28, 30, 31] and to the introduction of the papers [5, 13] and the references therein. The operator \(C_1\) was thoroughly investigated on weighted Banach spaces in [2]; see also [12]. Certain variants of the Cesàro operator \(C_1\) are considered in [9, 16].
Our aim is to investigate the operators \(C_t\), for \(t\in [0,1]\), when they are suitably interpreted to act on the space \(H({{\mathbb {D}}})\) of holomorphic functions on the open unit disc \({{\mathbb {D}}}:=\{z\in {{\mathbb {C}}}\,:\ |z|<1\}\), on the Banach space \(H^\infty \) of bounded analytic functions and on the weighted Banach spaces \(H_v^\infty \) and \(H_v^0\) with their sup-norms. The space \(H({{\mathbb {D}}})\) is equipped with the topology \(\tau _c\) of uniform convergence on the compact subsets of \({{\mathbb {D}}}\). According to [21, §27.3(3)] the space \(H({{\mathbb {D}}})\) is a Fréchet–Montel space. A family of norms generating \(\tau _c\) is given, for each \(0<r<1\), by
A weight v is a continuous, non-increasing function \(v:[0,1)\rightarrow (0,\infty )\). We extend v to \({{\mathbb {D}}}\) by setting \(v(z):=v(|z|)\), for \(z\in {{\mathbb {D}}}\). Note that \(v(z)\le v(0)\) for all \(z\in {{\mathbb {D}}}\). Given a weight v on [0, 1), we define the corresponding weighted Banach spaces of analytic functions on \({{\mathbb {D}}}\) by
and
both endowed with the norm \(\Vert \cdot \Vert _{\infty ,v}\). Since \(\Vert f\Vert _{\infty ,v}\le v(0)\Vert f\Vert _\infty \) whenever \(f\in H^\infty \), it is clear that \(H^\infty \subseteq H^\infty _v\) with a continuous inclusion. If \(v(z)=1\) for all \(z\in {{\mathbb {D}}}\), then \(H^\infty _v\) coincides with the space \(H^\infty \) of all bounded analytic functions on \({{\mathbb {D}}}\) with the sup-norm \(\Vert \cdot \Vert _\infty \) and \(H^0_v\) reduces to \(\{0\}\). Moreover, \(H^\infty _v\subseteq H({{\mathbb {D}}})\) continuously. Indeed, fix \(0<r<1\). Then \(\frac{1}{v(0)}\le \frac{1}{v(z)}\le \frac{1}{v(r)}\) for \(|z|\le r\) and so (1.4) implies that
We refer the reader to [10] for a recent survey of such types of weighted Banach spaces and operators between them.
Whenever necessary we will identify a function \(f\in H({{\mathbb {D}}})\) with its sequence of Taylor coefficients \({\hat{f}}:=({\hat{f}}(n))_{n\in {{\mathbb {N}}}_0}\) (i.e., \({\hat{f}}(n):=\frac{f^{(n)}(0)}{n!}\), for \(n\in {{\mathbb {N}}}_0\)), so that \(f(z)=\sum _{n=0}^\infty {\hat{f}}(n)z^n\), for \(z\in {{\mathbb {D}}}\). The linear map \(\Phi :H({{\mathbb {D}}})\rightarrow \omega \) is defined by
It is injective (clearly) and continuous. Indeed, for each \(m\in {{\mathbb {N}}}_0\),
is a continuous seminorm in \(\omega \). Fix \(0<r<1\), in which case
for each \(f\in H({{\mathbb {D}}})\) because \(\frac{1}{r^j}\le \frac{1}{r^m}\) for all \(0\le j\le m\). Of course, the increasing sequence of seminorms \(\{r_m\,\ m\in {{\mathbb {N}}}_0\}\) generates the topology of \(\omega \).
We first provide an integral representation of the generalized Cesàro operators \(C_t\) defined on \(H({{\mathbb {D}}})\), for \(t\in [0,1)\). So, fix \(t\in [0,1)\) and define \(C_t:H({{\mathbb {D}}})\rightarrow H({{\mathbb {D}}})\) by \( C_tf(0):=f(0)\) and
for every \(f\in H({{\mathbb {D}}})\). It turns out that \(C_t\) is continuous on \(H({{\mathbb {D}}})\); see Proposition 2.1. Moreover, the discrete Cesàro operator \(C_t:\omega \rightarrow \omega \), when restricted to the subspace \(\Phi (H({{\mathbb {D}}}))\subseteq \omega \) is transferred to \(H({{\mathbb {D}}})\) as follows. For a fixed \(f\in H({{\mathbb {D}}})\) we have \(f(\xi )=\sum _{n=0}^\infty a_n \xi ^n\), for \(\xi \in {{\mathbb {D}}}\), with \({\hat{f}}=(a_n)_{n\in {{\mathbb {N}}}_0}\) its sequence of Taylor coefficients. Since \(\frac{1}{1-t\xi }=\sum _{n=0}^\infty t^n\xi ^n\), for \(\xi \in {{\mathbb {D}}}\), we can form the Cauchy product of the two series, thereby obtaining
Then (1.5) yields
The interchange of the infinite sum and the integral is permissible by uniform convergence of the series. This shows that \(C_tf\in H({{\mathbb {D}}})\) also has the series representation
where the coefficients of the series are precisely as in (1.1). For the sake of clarity we will denote the discrete generalized Cesàro operator \(C_t:\omega \rightarrow \omega \) by \(C_t^\omega \) and reserve the notation \(C_t\) for the operator (1.5) acting in \(H({{\mathbb {D}}})\). Note that \(C_0^\omega =D_\varphi \) (see (1.2)). Moreover, \(C_0\) is given by \(C_0f(z)=\frac{1}{z}\int _0^z f(\xi )\,d\xi \) for \(z\not =0\) and \(C_0f(0)=f(0)\), which is the classical Hardy operator in \(H({{\mathbb {D}}})\).
The main results for \(C_t\) when acting in the Fréchet space \(H({{\mathbb {D}}})\) occur in Proposition 2.1 (continuity), Proposition 3.3 (non-compactness), Proposition 3.7 (spectra) and Proposition 3.8 (linear dynamics and mean ergodicity). For the analogous information concerning \(C_t\) when acting in the weighted Banach spaces \(H^\infty _v\) and \(H^0_v\) see Proposition 2.4 and Corollary 2.5 (continuity), Proposition 2.7 (compactness), Proposition 2.8 (spectra) and Proposition 3.2 (linear dynamics and mean ergodicity).
We end this section by recalling a few definitions and some notation concerning locally convex spaces and operators between them. For further details about functional analysis and operator theory relevant to this paper see, for example, [15, 18, 20,21,22, 29].
Given locally convex Haudorff spaces X, Y (briefly, lcHs) we denote by \({{\mathcal {L}}}(X,Y)\) the space of all continuous linear operators from X into Y. If \(X=Y\), then we simply write \({{\mathcal {L}}}(X)\) for \({{\mathcal {L}}}(X,X)\). Equipped with the topology of pointwise convergence on X (i.e., the strong operator topology) the lcHs \({{\mathcal {L}}}(X)\) is denoted by \({{\mathcal {L}}}_s(X)\). Equipped with the topology \(\tau _b\) of uniform convergence on the bounded subsets of X the lcHs \({{\mathcal {L}}}(X)\) is denoted by \({{\mathcal {L}}}_b(X)\).
Let X be a lcHs space. The identity operator on X is denoted by I. The transpose operator of \(T\in {{\mathcal {L}}}(X)\) is denoted by \(T'\); it acts from the topological dual space \(X':={{\mathcal {L}}}(X,{{\mathbb {C}}})\) of X into itself. Denote by \(X'_\sigma \) (resp., by \(X'_\beta \)) the topological dual \(X'\) equipped with the weak* topology \(\sigma (X',X)\) (resp., with the strong topology \(\beta (X',X)\)); see [21, §21.2] for the definition. It is known that \(T'\in {{\mathcal {L}}}(X'_\sigma )\) and \(T'\in {{\mathcal {L}}}(X_\beta ')\), [22, p. 134]. The bi-transpose operator \((T')'\) of T is simply denoted by \(T''\) and belongs to \({{\mathcal {L}}}((X'_\beta )'_\beta )\).
A linear map \(T:X\rightarrow Y\), with X, Y lcHs’, is called compact if there exists a neighbourhood \({{\mathcal {U}}}\) of 0 in X such that \(T({{\mathcal {U}}})\) is a relatively compact set in Y. It is routine to show that necessarily \(T\in {{\mathcal {L}}}(X,Y)\). We recall the following well known result; see [20, Proposition 17.1.1], [22, §42.1(1)].
Lemma 1.1
Let X be a lcHs. The compact operators are a 2-sided ideal in \({{\mathcal {L}}}(X)\).
Given a lcHs X 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 set \(\sigma (T;X):={{\mathbb {C}}}{\setminus } \rho (T;X)\) is called the spectrum of T. The point spectrum \(\sigma _{pt}(T;X)\) of T consists of all \(\lambda \in {{\mathbb {C}}}\) (also called an eigenvalue of T) such that \((\lambda I-T)\) is not injective. Some authors (eg. [29]) prefer the subset \(\rho ^*(T;X)\) of \(\rho (T;X)\) consisting of all \(\lambda \in {{\mathbb {C}}}\) for which there exists \(\delta >0\) such that the open disc \(B(\lambda ,\delta ):=\{z\in {{\mathbb {C}}}:\, |z-\lambda |<\delta \}\subseteq \rho (T;X)\) and \(\{R(\mu ,T):\, \mu \in B(\lambda ,\delta )\}\) is an equicontinuous subset of \({{\mathcal {L}}}(X)\). Define \(\sigma ^*(T;X):={{\mathbb {C}}}{\setminus } \rho ^*(T;X)\), which is a closed set with \(\sigma (T;X)\subseteq \sigma ^*(T;X)\). For the spectral theory of compact operators in lcHs’ we refer to [15, 18], for linear dynamics to [6, 17] and for mean ergodic operators to [23], for example.
2 Continuity, compactness and spectrum of \(C_t\)
In this section we establish, for \(t\in [0,1)\), the continuity of \(C_t:H({{\mathbb {D}}})\rightarrow H({{\mathbb {D}}})\) as well as the continuity of \(C_t\) from \(H^\infty \) (resp., \(H^\infty _v\)) into \(H^\infty \) (resp., \(H^\infty _v\)). The same is true for \(C_t:H^0_v\rightarrow H^0_v\) whenever \(\lim _{r\rightarrow 1^-}v(r)=0\). It is also shown that the bi-transpose \(C_t''\) of \(C_t\in {{\mathcal {L}}}(H^0_v)\) is the generalized Cesàro operator \(C_t\in {{\mathcal {L}}}(H^\infty _v)\), provided that \(\lim _{r\rightarrow 1^-}v(r)=0\). For such weights v it also turns out that both \(C_t\in {{\mathcal {L}}}(H^0_v)\) and \(C_t\in {{\mathcal {L}}}(H^\infty _v)\) are compact operators (cf. Proposition 2.7); their spectrum is identified in Proposition 2.8. Of particular interest are the standard weights \(v_\gamma (z):=(1-|z|)^\gamma \), for \(\gamma >0\) and \(z\in {{\mathbb {D}}}\).
Proposition 2.1
For every \(t\in [0,1)\) the operator \(C_t:H({{\mathbb {D}}})\rightarrow H({{\mathbb {D}}})\) is continuous. Moreover, the set \(\{C_t:\, t\in [0,1)\}\) is equicontinuous in \({{\mathcal {L}}}(H({{\mathbb {D}}}))\).
Proof
Fix \(f\in H({{\mathbb {D}}})\). Taking into account that \(C_tf(0)=f(0)\), for all \(t\in [0,1)\) and, for each \(r\in (0,1)\), that \(\sup _{|z|\le r}|C_tf(z)|=\sup _{|z|=r}|C_tf(z)|\), the formula (1.5) implies, for each \(z\in {{\mathbb {D}}}{\setminus }\{0\}\), that
because \(|1-t\xi |\ge 1-t|\xi |\ge 1-|\xi |\ge 1-|z|\), for all \(|\xi |\le |z|\). It follows from the previous inequality, for each \(r\in (0,1)\), that
see (1.4). This implies the result. \(\square \)
The following example will prove to be useful in the sequel.
Example 2.2
Consider the constant function \(f_1(z):=1\), for every \(z\in {{\mathbb {D}}}\), in which case \(C_tf_1(0)=f_1(0)=1\) for every \(t\in [0,1]\). For \(t=0\), it was noted in Sect. 1 that \(C_0\) is
the Hardy operator. In particular, \(C_0f_1(z)=1\), for every \(z\in {{\mathbb {D}}}\). For \(t\in (0,1]\), note that \(C_tf_1(0)=1\) and
For \(t=1\) this shows, in particular, that \(C_1(H^\infty )\not \subset H^\infty \), which is well known. For an investigation of the operator \(C_1\) acting in \(H^\infty \) we refer to [14].
Concerning \(t\in (0,1)\), recall the Taylor series expansion
from which it follows that
with the series having radius of convergence \(\frac{1}{t}>1\). The claim is that \(\Vert C_tf_1\Vert _\infty =\sup _{|z|<1}|C_tf_1(z)|=-\frac{\log (1-t)}{t}\). Indeed, \(C_tf_1\) is clearly holomorhic in \(B(0,\frac{1}{t}):=\{\xi \in {{\mathbb {C}}}:\, |\xi |<\frac{1}{t}\}\) hence, continuous in \(B(0,\frac{1}{t})\), and satisfies \(C_tf_1(1)=-\frac{\log (1-t)}{t}\) with \(\lim _{r\rightarrow 1^-}C_tf_1(r)=C_tf_1(1)\). On the other hand, for every \(z\in {{\mathbb {D}}}{\setminus }\{0\}\) and \(t\in (0,1)\) we have that
This completes the proof of the claim. Observe that \(\Vert C_tf_1\Vert _\infty >1\). Indeed, define \(\gamma (t)=-\log (1-t)-t\), for \(t\in [0,1)\). Then \(\gamma (0)=0\), \(\lim _{t\rightarrow 1^-}\gamma (t)=\infty \) and \(\gamma '(t)=\frac{1}{1-t}-1=\frac{t}{1-t}\), for \(t\in [0,1)\). Since \(\gamma '(t)>0\), for \(t\in (0,1)\), it follows that \(\gamma \) is strictly increasing and so \(\gamma (t)>0\) for all \(t\in (0,1)\). This implies that \(\Vert C_tf_1\Vert _\infty =-\frac{\log (1-t)}{t}>1\) for every \(t\in (0,1)\). On the other hand, for \(t\in (0,1)\), the inequality \(\sum _{n=0}^\infty t^n/(n+1)<\sum _{n=0}^\infty t^n\) implies that \(-\frac{\log (1-t)}{t}<\frac{1}{1-t}\). So, we have shown that \(\Vert C_0f_1\Vert _\infty =1\) and
We now turn to the action of \(C_t\) in various Banach spaces. For \(t=1\) it was noted above that \(C_1\) fails to act in \(H^\infty \).
Proposition 2.3
For \(t\in [0,1)\) the operator \(C_t:H^\infty \rightarrow H^\infty \) is continuous. Moreover, \(\Vert C_0\Vert _{H^\infty \rightarrow H^\infty }=1\) and
Proof
Let \(f\in H^\infty \) be fixed. Then
This implies that \(\Vert C_0\Vert _{H^\infty \rightarrow H^\infty }\le 1\). On the other hand, \(C_0f_1=f_1\) and so we can conclude that \(\Vert C_0\Vert _{H^\infty \rightarrow H^\infty }=1\).
Now let \(t\in (0,1)\). Then, for the parametrization \(\xi :=sz\), for \(s\in (0,1)\), it follows from \(|1-stz|\ge 1-|stz|\ge 1-st\) that
So, \(C_t\in {{\mathcal {L}}}(H^\infty )\) with \(\Vert C_t\Vert _{H^\infty \rightarrow H^\infty }\le -\frac{\log (1-t)}{t}\). But, \(\Vert C_tf_1\Vert _\infty = -\frac{\log (1-t)}{t}\). Accordingly, \(\Vert C_t\Vert _{H^\infty \rightarrow H^\infty }= -\frac{\log (1-t)}{t}\). \(\square \)
Proposition 2.4
Let v be a weight function on [0, 1). For each \(t\in [0,1)\) the operator \(C_t:H_v^\infty \rightarrow H_v^\infty \) is continuous. Moreover, \(\Vert C_0\Vert _{H^\infty _v\rightarrow H_v^\infty }=1\) and
Proof
Recall that \(C_tf(0):=f(0)\) for each \(f\in H({{\mathbb {D}}})\) and \(t\in [0,1]\). Fix \(t\in (0,1)\). Given \(f\in H^\infty _v\) and \(z\in {{\mathbb {D}}}{\setminus }\{0\}\), observe that
where we used that \(v(sz)=v(s|z|)\ge v(|z|)=v(z)\), for \(s\in (0,1)\), as v is non-increasing on (0, 1) and that \(|1-stz|\ge 1-st|z|\), for \(s\in (0,1)\). According to the calculations in Example 2.2 we can conclude that
This implies that \(C_t\in {{\mathcal {L}}}(H^\infty _v)\) and \(\Vert C_t\Vert _{H^\infty _v\rightarrow H^\infty _v}\le -\frac{\log (1-t)}{t}\).
For \(t=0\) observe that
as \(v(\xi )=v(z)\) whenever \(|\xi |=|z|\) with \(\xi \in {{\mathbb {D}}}\). This shows that \(\Vert C_0\Vert _{H^\infty _v\rightarrow H_v^\infty }\le 1\). Since \(C_0f_1=f_1\), it follows that actually \(\Vert C_0\Vert _{H^\infty _v\rightarrow H_v^\infty }=1\).
It remains to show that \(\Vert C_t\Vert _{H^\infty _v\rightarrow H^\infty _v}\ge 1\) for \(t\in (0,1)\). To this end, fix \(t\in (0,1)\) and consider the function \(g_0(z):=\frac{1}{1-tz}=\sum _{n=0}^\infty t^nz^n\), for \(z\in {{\mathbb {D}}}\). Then \(\Vert g_0\Vert _\infty =\frac{1}{1-t}\) and so \(g_0\in H^\infty \subseteq H^\infty _v\). Moreover, for every \(z\in {{\mathbb {D}}}{\setminus }\{0\}\), it is the case that
It follows that \(\Vert g_0\Vert _{\infty ,v}=\Vert C_tg_0\Vert _{\infty ,v}\le \Vert C_t\Vert _{H^\infty _v\rightarrow H^\infty _v}\Vert g_0\Vert _{\infty ,v}\) which implies that \(\Vert C_t\Vert _{H^\infty _v\rightarrow H^\infty _v}\ge 1\). \(\square \)
Corollary 2.5
Let v be a weight function on [0, 1) satisfying \(\lim _{r\rightarrow 1^-}v(r)=0\). For each \(t\in [0,1)\) the operator \(C_t:H^0_v\rightarrow H^0_v\) is continuous and satisfies \(\Vert C_t\Vert _{H^0_v\rightarrow H^0_v}=\Vert C_t\Vert _{H^\infty _v\rightarrow H^\infty _v}\).
Proof
By Proposition 2.4 and the fact that \(H^0_v\) is a closed subspace of \(H^\infty _v\), to obtain the result it suffices to establish that \(C_t(H_v^0)\subseteq H^0_v\). To this effect, observe that \(H^\infty \subseteq H^0_v\) and that \(H^\infty \) is dense in \(H^0_v\), as the space of polynomials is dense in \(H^0_v\); see Section 1 of [11] and also [7]. Proposition 2.3 implies that \(C_t(H^\infty )\subseteq H^\infty \subseteq H^0_v\). Since \(C_t\) acts continuously on \(H^\infty _v\), it follows that
Moreover, \(\lim _{r\rightarrow 1^-}v(r)=0\) implies that \(H^\infty _v\) is canonically isometric to the bidual of \(H^0_v\), [8, Example 2.1], and that the bi-transpose \(C_t'':H^\infty _v\rightarrow H_v^\infty \) of \(C_t:H^0_v\rightarrow H^0_v\) coincides with \(C_t:H^\infty _v\rightarrow H^\infty _v\) (see Lemma 2.6 below), from which the identity \(\Vert C_t\Vert _{H^0_v\rightarrow H^0_v}=\Vert C_t\Vert _{H^\infty _v\rightarrow H^\infty _v}\) follows. \(\square \)
Lemma 2.6
Let v be a weight function on [0, 1) satisfying \(\lim _{r\rightarrow 1^-}v(r)=0\). For each \(t\in [0,1)\), the bi-transpose \(C''_t:H^\infty _v\rightarrow H^\infty _v\) of \(C_t:H^0_v\rightarrow H_v^0\) coincides with \(C_t:H^\infty _v\rightarrow H^\infty _v\).
Proof
By Proposition 2.3 and Corollary 2.5, together with the fact that \(H^\infty _v\) is canonically isometric to the bidual of \(H^0_v\), both of the operators \(C_t''\) and \(C_t\) act continuously on \(H^\infty _v\).
To show that the bi-transpose \(C_t'':H^\infty _v\rightarrow H^\infty _v\) of \(C_t:H^0_v\rightarrow H^0_v\) coincides with \(C_t:H^\infty _v\rightarrow H^\infty _v\) we proceed via several steps.
First step Given \(f\in H({{\mathbb {D}}})\), its Taylor polynomials \(p_k(z)=\sum _{j=0}^k{\hat{f}}(j)z^j\), \(z\in {{\mathbb {D}}}\), for \(k\in {{\mathbb {N}}}_0\), converge to f uniformly on compact subsets of \({{\mathbb {D}}}\). That is, \(p_k\rightarrow f\) in \((H({{\mathbb {D}}}),\tau _c)\) as \(k\rightarrow \infty \). Accordingly, the averages of \((p_k)_{k\in {{\mathbb {N}}}_0}\), that is, the Cesàro means \(f_n(z):=\frac{1}{n+1}\sum _{j=0}^np_j(z)\), for \(z\in {{\mathbb {D}}}\) and \(n\in {{\mathbb {N}}}_0\), also converge to f in \((H({{\mathbb {D}}}),\tau _c)\) as \(n\rightarrow \infty \).
Second step Lemma 1.1 in [7] implies, for every \(f\in H^\infty _v\) and \(n\in {{\mathbb {N}}}_0\), that \(\Vert f_n\Vert _{\infty ,v}\le \Vert f\Vert _{\infty ,v}\), where \(f_n\) is the n-th Cesàro mean of f, as defined in the First step. Denote by \(U_v\) the closed unit ball of \((H^\infty _v,\Vert \cdot \Vert _{\infty ,v})\). Then, for any given \(f\in U_v\), its sequence of Cesàro means satisfies \((f_n)_{n\in {{\mathbb {N}}}_0}\subseteq U_v\) and \(f_n\rightarrow f\) in \((H({{\mathbb {D}}}),\tau _c)\) as \(n\rightarrow \infty \).
Third step With the topology of uniform convergence on the compact subsets of \(U_v\) denoted by \(\tau _c\), let \(X:=\{F\in (H_v^\infty )':\ F|_{U_v}\ \text { is } \tau _c-\text { continuous }\}\) be endowed with the norm \(\Vert F\Vert :=\sup \{|F(f)|:\ f\in U_v\}\). Then [8, Theorem 1.1(a)] ensures that \((X,\Vert \cdot \Vert )\) is a Banach space and that the evaluation map \(\Psi :H^\infty _v\rightarrow X'\) defined by \((\Psi (f))(F):=\langle f,F\rangle \), for \(F\in X\) and \(f\in H^\infty _v\), is an isometric isomorphism onto \(X'\) (where \(X'\) is the dual Banach space of \((X,\Vert \cdot \Vert )\)). Moreover, by [8, Theorem 1.1(b) and Example 2.1] the restriction map \(R:X\rightarrow (H^0_v)'\) given by \(F\mapsto F|_{H_v^0}\), is also a surjective isometric isomorphism. Therefore, the spaces \(H^\infty _v\) and \((H^0_v)''\) are isometrically isomorphic, that is, X and \((H^0_v)'\) are isometrically isomorphic and hence, also \(H^\infty _v\) and \((H^0_v)''\) are isometrically isomorphic.
It is easy to see, since the Banach space X above is the predual of \(H^\infty _v\), that the evaluation map \(\delta _z\in X\), for every \(z\in {{\mathbb {D}}}\), where \(\delta _z:f\mapsto f(z)\), for \(f\in H^\infty _v\), satisfies \(|\langle f,\delta _z\rangle |\le \Vert f\Vert _{\infty ,v}/v(z)\). In particular, the linear span L of the set \(\{\delta _z:\ z\in {{\mathbb {D}}}\}\) separates the points of \(H^\infty _v=X'\) and hence, L is dense in X. Therefore, the pointwise convergence topology \(\tau _p\) on \(H^\infty _v\) is Hausdorff and coarser than the \(w^*\)-topology \(\sigma (H^\infty _v, X)\).
Fourth step The closed unit ball \(U_v\) of \(H^\infty _v\) is a \(\tau _c\)-compact set by Montel’s theorem, as it is \(\tau _c\)-bounded and closed. On the other hand, \(U_v\) is also \(\sigma (H^\infty _v,X)\)-compact by the Alaoglu-Bourbaki theorem. Since \(\tau _p|_{U_v}\) is coarser than \(\tau _c|_{U_v}\) and Hausdorff, we can conclude that \(\tau _p|_{U_v}=\tau _c|_{U_v}\). In the same way, it follows that \(\tau _p|_{U_v}=\sigma (H^\infty _v,X)|_{U_v}\). Accordingly, \(\tau _p|_{U_v}=\tau _c|_{U_v}=\sigma (H^\infty _v,X)|_{U_v}\).
We are now ready to prove that \((C_t)''=C_t\). To show this, it suffices to establish that \((C_t)''f=C_tf\) for every \(f\in U_v\).
So, fix \(f\in U_v\). With \((f_n)_{n\in {{\mathbb {N}}}_0}\) as in the First step it follows from there that \(f_n\rightarrow f\) in \((H({{\mathbb {D}}}),\tau _c)\) as \(n\rightarrow \infty \) and, by the Second step, that \((f_n)_{n\in {{\mathbb {N}}}_0}\subseteq U_v\). This implies that \(C_tf_n\rightarrow C_tf\) in \((H({{\mathbb {D}}}),\tau _c)\) as \(n\rightarrow \infty \). Since \(C_t\in {{\mathcal {L}}}(H^\infty _v)\) and \(f\in U_v\), it is clear that \(C_tf\in H^\infty _v\). On the other hand, by the Fourth step the sequence \((f_n)_{n\in {{\mathbb {N}}}_0}\) also converges to f in \((H^\infty _v,\sigma (H^\infty _v,X))=(H^\infty _v,\sigma (H^\infty _v, (H^0_v)'))\). Since \((C_t)'':((H^0_v)'',\sigma ((H^0_v)'',(H^0_v)'))\rightarrow ((H^0_v)'',\sigma ((H^0_v)'',(H^0_v)')\) is continuous, [20, §8.6], that is, \((C_t)'':(H^\infty _v,\sigma (H^\infty _v, X))\rightarrow (H^\infty _v,\sigma (H^\infty _v, X))\) is continuous, it follows that \((C_t)''f_n\rightarrow (C_t)''f\) in \((H^\infty _v,\sigma (H^\infty _v, X))\) as \(n\rightarrow \infty \). Now, \((f_n)_{n\in {{\mathbb {N}}}_0}\subset H^\infty \subseteq H^0_v\), as each \(f_n\) is a polynomial, and \((C_t)''f_n=C_tf_n\) for every \(n\in {{\mathbb {N}}}_0\). Moreover, the sequence \(C_tf_n\rightarrow (C_t)''f\) in \((H({{\mathbb {D}}}),\tau _p)\) as \(n\rightarrow \infty \). Thus, \((C_t)''f=C_tf\) as desired. \(\square \)
Proposition 2.7
Let v be a weight function satisfying \(\lim _{r\rightarrow 1^-}v(r)=0\). For each \(t\in [0,1)\), both of the operators \(C_t:H^\infty _v\rightarrow H^\infty _v\) and \(C_t\rightarrow H^0_v\rightarrow H^0_v\) are compact.
Proof
Fix \(t\in [0,1)\). Since \(H^0_v\) is a closed subspace of \(H_v^\infty \) and \(C_t(H^0_v)\subseteq H_v^0\) (cf. Corollary 2.5), it suffices to show that \(C_t:H^\infty _v\rightarrow H_v^\infty \) is compact. First we establish the following Claim:
-
(*)
Let the sequence \((f_n)_{n\in {{\mathbb {N}}}}\subset H^\infty _v\) satisfy \(\Vert f_n\Vert _{\infty ,v}\le 1\) for every \(n\in {{\mathbb {N}}}\) and \(f_n\rightarrow 0\) in \((H({{\mathbb {D}}}),\tau _{c})\) for \(n\rightarrow \infty \). Then \(C_tf_n\rightarrow 0\) in \(H^\infty _v\).
To prove the Claim, let \((f_n)_{n\in {{\mathbb {N}}}}\subset H^\infty _v\) be a sequence as in (*). Fix \(\varepsilon >0\) and select \(\delta \in (0,\beta )\), where \(\beta :=\min \{1,\frac{\varepsilon (1-t)}{2},\frac{\varepsilon (1-t)}{2v(0)}\}\). Since \(\{\xi \in {{\mathbb {C}}}\ \ |\xi |\le (1-\delta )\}\) is a compact subset of \({{\mathbb {D}}}\), there exists \(n_0\in {{\mathbb {N}}}\) such that
Recall that \(C_tf_n(0)=f_n(0)\) for every \(n\in {{\mathbb {N}}}\). For \(z\in {{\mathbb {D}}}{\setminus }\{0\}\) we have seen previously that
Denote the first (resp., second) summand in the right-side of the previous inequality by \((A_n)\) (resp., by \((B_n)\)). Using the facts that \(|1-stz|\ge 1-st |z|\ge \max \{1-s,1-t,1-|z|\}\), for all \(s,t\in [0,1)\) and \(z\in {{\mathbb {D}}}\), and that v is non-increasing on [0, 1) it follows, for every \(n\ge n_0\), that \(\int _0^{1-\delta }|f_n(sz)|\,ds\le (1-\delta )\max _{|\xi |\le (1-\delta )}|f_n(\xi )|\) (as \(|sz|\le (1-\delta )\) for all \(s\in [0,1-\delta ]\)) and hence, that
On the other hand, for every \(n\ge n_0\), we have (as \(\Vert f_n\Vert _{\infty ,v}=\sup _{\xi \in {{\mathbb {D}}}}v(\xi )|f_n(\xi )|\le 1\)) that
It follows that \(\Vert C_tf_n\Vert _{\infty ,v}<\varepsilon \) for every \(n\ge n_0\). That is, \(C_tf_n\rightarrow 0\) in \(H^\infty _v\) for \(n\rightarrow \infty \) and so (*) is proved.
The compactness of \(C_t\in {{\mathcal {L}}}(H^\infty _v)\) can be deduced from (*) as follows. Let \((f_n)_{n\in {{\mathbb {N}}}}\subset H^\infty _v\) be any bounded sequence. There is no loss of generality in assuming that \(\Vert f_n\Vert _{\infty ,v}\le 1\) for all \(n\in {{\mathbb {N}}}\). To establish the compactness of \(C_t\in {{\mathcal {L}}}(H^\infty _v)\) we need to show that \((C_tf_n)_{n\in {{\mathbb {N}}}}\) has a convergent subsequence in \(H^\infty _v\).
Since \(H^\infty _v\subseteq H({{\mathbb {D}}})\) continuously, the sequence \((f_n)_{n\in {{\mathbb {N}}}}\) is also bounded in the Fréchet–Montel space \(H({{\mathbb {D}}})\). Hence, there is a subsequence \(g_j:=f_{n_j}\), for \(j\in {{\mathbb {N}}}\), of \((f_n)_{n\in {{\mathbb {N}}}}\) and \(f\in H({{\mathbb {D}}})\) such that \(g_j\rightarrow f\) in \(H({{\mathbb {D}}})\) with respect to \(\tau _c\). In particular, \(g_j\rightarrow f\) pointwise on \({{\mathbb {D}}}\). Since \(v(z)|g_j(z)|=v(z)|f_{n_j}(z)|\le 1\) for all \(z\in {{\mathbb {D}}}\) and \(j\in {{\mathbb {N}}}\), letting \(j\rightarrow \infty \) it follows that \(v(z)|f(z)|\le 1\) for all \(z\in {{\mathbb {D}}}\), that is, \(f\in H^\infty _v\) with \(\Vert f\Vert _{\infty ,v}\le 1\). Let \(h_j:=\frac{1}{2}(g_j-f)\), for \(j\in {{\mathbb {N}}}\). Then \(\Vert h_j\Vert _{\infty , v}\le 1\), for \(j\in {{\mathbb {N}}}\), and \(h_j\rightarrow 0\) in \(H({{\mathbb {D}}})\) with respect to \(\tau _c\). Condition (*) implies that \(C_th_j\rightarrow 0\) in \(H^\infty _v\) from which it follows that \(C_tf_{n_j}=C_tg_j=C_t(g_j-f)+C_tf=2C_th_j+C_tf\rightarrow C_tf \) in \(H^\infty _v\), as desired. \(\square \)
Proposition 2.8
Let v be a weight function on [0, 1) satisfying \(\lim _{r\rightarrow 1^-}v(r)=0\). For each \(t\in [0,1)\) the spectra of \(C_t\in {{\mathcal {L}}}(H^\infty _v)\) and of \(C_t\in {{\mathcal {L}}}(H^0_v)\) are given by
and
Proof
Let \(t\in [0,1)\) be fixed. By [13, Lemma 3.6] we know that the point spectrum of the operator \(C_t^\omega \in {{\mathcal {L}}}(\omega )\) is given by \(\sigma _{pt}(C_t^\omega ;\omega )=\{\frac{1}{m+1}:\, m\in {{\mathbb {N}}}_0\}\) and, for each \(m\in {{\mathbb {N}}}_0\), that the corresponding eigenspace \(\textrm{Ker}(\frac{1}{m+1}I-C_t^\omega )\) is 1-dimensional and is generated by an eigenvector \(x^{[m]}=(x_n^{[m]})_{n\in {{\mathbb {N}}}_0}\in \ell ^1\). Since \(H^0_v\subseteq H^\infty _v\subseteq H({{\mathbb {D}}})\) with continuous inclusions and \(\Phi :H({{\mathbb {D}}})\rightarrow \omega \) (cf. Sect. 1) is a continuous embedding, this implies that \(\sigma _{pt}(C_t;H^0_v)\subseteq \sigma _{pt}(C_t;H^\infty _v)\subseteq \{\frac{1}{m+1}\,:\, m\in {{\mathbb {N}}}_0\}\). Indeed, let \(f\in H({{\mathbb {D}}}){\setminus }\{0\}\) and \(\lambda \in {{\mathbb {C}}}\) satisfy \(C_tf=\lambda f\). Then \(\lambda f(z)=\sum _{n=0}^\infty \widehat{(\lambda f)}(n)z^n=\sum _{n=0}^\infty \lambda {\hat{f}}(n)z^n\) and, by (1.6), we have that \((C_tf)(z)=\sum _{n=0}^\infty (C_t^\omega {\hat{f}})_nz^n\). It follows that \(C_t^\omega {\hat{f}}=\lambda {\hat{f}}\) in \(\omega \) with \({\hat{f}}\not =0\) and so \(\lambda \in \sigma _{pt}(C_t^\omega ;\omega )=\{\frac{1}{m+1}\,\ m\in {{\mathbb {N}}}_0\}\).
To conclude the proof, it remains to show that \(\{\frac{1}{m+1}\,:\, m\in {{\mathbb {N}}}_0\}\subseteq \sigma _{pt}(C_t;H^0_v)\). To establish this recall, for each \(m\in {{\mathbb {N}}}_0\), that the eigenvector \(x^{[m]}\in \ell ^1\) and hence, the function \(g_m(z):=\sum _{n=0}^\infty (x^{[m]})_nz^n\) belongs to \(H^0_v\) because \(0\le v(z)|g_m(z)|\le v(z)\Vert x^{[m]}\Vert _{\ell ^1}\) for \(z\in {{\mathbb {D}}}\) and \(\lim _{r\rightarrow 1^-}v(r)=0\). Moreover, according to (1.5) and (1.6) we have, for each \(z\in {{\mathbb {D}}}\), that
Thus \(g_m\) is an eigenvector of \(C_t\in {{\mathcal {L}}}(H^0_v)\) corresponding to the eigenvalue \(\frac{1}{m+1}\).
The validity of \(\sigma (C_t;H^0_v)=\sigma (C_t;H^\infty _v)=\{\frac{1}{m+1}\,:\, m\in {{\mathbb {N}}}_0\}\cup \{0\}\) follows from the fact that \(C_t\) is a compact operator on both spaces. \(\square \)
We now investigate the norm of \(C_t\) on \(H^\infty _v\) for the standard weights \(v_\gamma (z):=(1-|z|)^\gamma \), for \(\gamma >0\) and \(z\in {{\mathbb {D}}}\), which satisfy \(\lim _{r\rightarrow 1^-}v_\gamma (r)=0\).
Proposition 2.9
Let \(t\in (0,1)\) and \(\gamma >0\).
-
(i)
The operator norm \(\Vert C_t\Vert _{H^\infty _{v_\gamma }\rightarrow H^\infty _{v_\gamma }}=1\), for every \(\gamma \ge 1\).
-
(ii)
For each \(\gamma \in (0,1)\), the inequality \(\Vert C_t\Vert _{H^\infty _{v_\gamma }\rightarrow H^\infty _{v_\gamma }}\le \min \{-\frac{\log (1-t)}{t},\frac{1}{\gamma }\}\) is valid.
Proof
We adapt the arguments given for the Cesàro operator \(C_1\) in the proof of [2, Theorem 2.3].
Let \(\gamma >0\) and \(t\in (0,1)\) be fixed. For \(f\in H^\infty _{v_\gamma }\) with \(\Vert f\Vert _{\infty ,v_\gamma }=1\) we have
as \(z\in {{\mathbb {D}}}\) implies that \(1-st|z|\ge 1-s|z|\), for \(s\in (0,1)\). Accordingly,
and hence,
Define \(\phi (s):=\frac{1-(1-s)^\gamma }{s}\) for \(s\in (0,1]\) and \(\phi (0)=\gamma \), in which case \(\phi \) is continuous. So, the previous inequality yields \(\Vert C_tf\Vert _{\infty ,v_\gamma }\le \frac{M_\gamma }{\gamma }\), for all \(\Vert f\Vert _{\infty ,v_\gamma }\le 1\), that is, \(\Vert C_t\Vert _{H^\infty _{v_\gamma }\rightarrow H^\infty _{v_\gamma }}\le \frac{M_\gamma }{\gamma }\), where \(M_\gamma :=\sup _{s\in [0,1]}\phi (s)\). Proposition 2.4 yields that \(1\le \Vert C_t\Vert _{H^\infty _{v_\gamma }\rightarrow H^\infty _{v_\gamma }}\le -\frac{\log (1-t)}{t}\) for \(t\in (0,1)\). On page 101 of [2] it is shown that \(\frac{M_\gamma }{\gamma }\le 1\) whenever \(\gamma \ge 1\) and that \(M_\gamma \le 1\) for all \(\gamma \in (0,1)\). The proof of both parts (i) and (ii) follows immediately. \(\square \)
Remark 2.10
For each \(\gamma >0\) let \(v_\gamma (z)=(1-|z|)^\gamma \), for \(z\in {{\mathbb {D}}}\). Proposition 2.9 implies that \(\sup _{0\le t<1}\Vert C_t\Vert _{H^\infty _{v_\gamma }\rightarrow H^\infty _{v_\gamma }}<\infty \). Moreover, if \(\gamma \ge 1\), then \(\Vert C_t^n\Vert _{H^\infty _{v_\gamma }\rightarrow H^\infty _{v_\gamma }}=1\) for every \(n\in {{\mathbb {N}}}\); see case (i) in the proof of [2, Theorem 2.3] together with the fact that \(1\in \sigma _{pt}(C_t, H^\infty _{v_\gamma })\) by Proposition 2.8.
Let \(n\in {{\mathbb {N}}}\) be fixed. Consider the weight \(v(z)=(\log \frac{e}{1-|z|})^{-n}\), for \(z\in {{\mathbb {D}}}\), which satisfies \(v(0)=1\) and \(\lim _{|z|\rightarrow 1^-}v(z)=0\).
The function \(f(z):=[\log (1-z)]^n\in H({{\mathbb {D}}})\) belongs to \(H^\infty _v\). Indeed, for each \(z\in {{\mathbb {D}}}\), we have that
and hence, that \(|f(z)|=|\log (1-z)|^n\le (-\log (1-|z|))^n\). Since v is given by \(v(z)=(1-\log (1-|z|))^{-n}\) and \(\lim _{|z|\rightarrow 1^-}\frac{-\log (1-|z|)}{1-\log (1-|z|)}=1\), it follows that \(\Vert f\Vert _{\infty ,v}<\infty \) and so \(f\in H^\infty _v\). On the other hand,
Accordingly, \(C_1f\not \in H^\infty _v\) since
This implies that the Cesàro operator \(C_1\) is not well-defined on \(H^\infty _v\), that is, \(C_1(H^\infty _v)\not \subseteq H^\infty _v\). But, by Proposition 2.4 the generalized Cesàro operator \(C_t\in {{\mathcal {L}}}(H^\infty _v)\) for every \(t\in [0,1)\). At this point, the following question arises: Is \(\sup _{t\in [0,1)}\Vert C_t\Vert _{H^\infty _v\rightarrow H^\infty _v}<\infty \) for this particular v? Our next two results show that the answer is negative for certain weights v, which includes \(v(z)=\left( \log \frac{e}{1-|z|}\right) ^{-n}\) for \(z\in {{\mathbb {D}}}\).
Proposition 2.11
Let v be a weight function on [0, 1) such that \(\sup _{t\in [0,1)}\Vert C_t\Vert _{H^\infty _v\rightarrow H^\infty _v}<\infty \). Then \(C_1\in {{\mathcal {L}}}(H^\infty _v)\).
Proof
Proposition 2.1 implies that \(\{C_t:\, t\in [0,1)\}\) is equicontinuous in \({{\mathcal {L}}}(H({{\mathbb {D}}}))\). The claim is that \(\lim _{t\rightarrow 1^-}C_tf(z)=C_1f(z)\), for every \(f\in H({{\mathbb {D}}})\) and \(z\in {{\mathbb {D}}}\).
To prove this claim fix \(f\in H({{\mathbb {D}}})\) and \(z\in {{\mathbb {D}}}{\setminus }\{0\}\). Recall, for \(t\in [0,1)\), that
and
Moreover, for each \(z\in {{\mathbb {D}}}{\setminus }\{0\}\), we have (as \(|1-stz|\ge (1-|z|)\)) that
and that \(\lim _{t\rightarrow 1^-}\frac{f(sz)}{1-st z}=\frac{f(sz)}{1-s z}\) for every \(s\in [0,1]\). So, we can apply the dominated convergence theorem to conclude that \(\lim _{t\rightarrow 1^-}C_tf(z)=C_1f(z)\) for \(z\in {{\mathbb {D}}}{\setminus }\{0\}\). For \(z=0\) we have \(C_tf(0)=f(0)=C_1f(0)\) for each \(f\in H({{\mathbb {D}}})\) and \(t\in [0,1)\). So, for each \(f\in H({{\mathbb {D}}})\), we can conclude that \(C_tf\rightarrow C_1f\) pointwise on \({{\mathbb {D}}}\) for \(t\rightarrow 1^-\). The claim is thereby established.
We now show that \(C_tf\rightarrow C_1f\) in \(H({{\mathbb {D}}})\) as \(t\rightarrow 1^-\) for every \(f\in H^\infty _v\). The assumption \(\sup _{t\in [0,1)}\Vert C_t\Vert _{H^\infty _v\rightarrow H_v^\infty }<\infty \) implies that there exists \(M>0\) satisfying \(\Vert C_t\Vert _{H^\infty _v\rightarrow H^\infty _v}\le M\) for every \(t\in [0,1)\). Therefore,
Fix \(f\in H^\infty _v\). Then \(\{C_tf:\, t\in [0,1)\}\) is a bounded set in \(H({{\mathbb {D}}})\). Indeed, given \(r\in (0,1)\) and \(t\in [0,1)\) we have (as \(v(r)\le v(z)\) for all \(|z|\le r\)) that
So, the set \(\{C_tf:\, t\in [0,1)\}\) is bounded in the Fréchet–Montel space \(H({{\mathbb {D}}})\) and hence, it is relatively compact in \(H({{\mathbb {D}}})\). Since \(C_tf\rightarrow C_1f\) pointwise on \({{\mathbb {D}}}\) for \(t\rightarrow 1^-\), it follows that \(C_tf\rightarrow C_1f\) with respect to \(\tau _{c}\), that is, in the Fréchet space \(H({{\mathbb {D}}})\), for \(t\rightarrow 1^-\). In particular, \(C_1f\in H({{\mathbb {D}}})\).
Since \(H^\infty _v\subseteq H({{\mathbb {D}}})\) and \(C_th\rightarrow C_1h\) pointwise on \({{\mathbb {D}}}\) as \(t\rightarrow 1^-\), for every \(h\in H({{\mathbb {D}}})\), letting \(t\rightarrow 1^-\) in (2.3) it follows that
that is, \(\Vert C_1f\Vert _{\infty ,v}\le M\Vert f\Vert _{\infty ,v}\). But, \(f\in H^\infty _v\) is arbitrary and so \(C_1\in {{\mathcal {L}}}(H^\infty _v)\). \(\square \)
Proposition 2.12
For each \(n\in {{\mathbb {N}}}\), let \(v(z)=(\log (\frac{e}{1-|z|}))^{-n}\) for \(z\in {{\mathbb {D}}}\). Then \(\sup _{t\in [0,1)}\Vert C_t\Vert _{H^\infty _v\rightarrow H^\infty _v}=\infty \).
Proof
Apply Proposition 2.11 and the discussion prior it. \(\square \)
3 Linear dynamics and mean ergodicity of \(C_t\)
The aim of this section is to investigate the mean ergodicity and the linear dynamics of the operators \(C_t\), for \(t\in [0,1)\), acting on \(H({{\mathbb {D}}})\), \(H^\infty _v\) and \(H^0_v\)
An operator \(T\in {{\mathcal {L}}}(X)\), with X a lcHs, is called power bounded if \(\{T^n:\ n\in {{\mathbb {N}}}_0\}\) is an equicontinuous subset of \({{\mathcal {L}}}(X)\). For a Banach space X, this means that \(\sup _{n\in {{\mathbb {N}}}_0}\Vert T^n\Vert _{X\rightarrow X}<\infty \). Given \(T\in {{\mathcal {L}}}(X)\), the averages
are usually called the Cesàro means of T. The operator T is said to be mean ergodic (resp., uniformly mean ergodic) if \((T_{[n]})_{n\in {{\mathbb {N}}}}\) is a convergent sequence in \({{\mathcal {L}}}_s(X)\) (resp., in \({{\mathcal {L}}}_b(X)\)). It is routine to check that \(\frac{T^n}{n}=T_{[n]}-\frac{n-1}{n}T_{[n-1]}\), for \(n\ge 2\), and hence, \(\tau _s\)-\(\lim _{n\rightarrow \infty }\frac{T^n}{n}=0\) whenever T is mean ergodic. Every power bounded operator on a Fréchet–Montel space X is necessarily uniformly mean ergodic, [1, Proposition 2.8]. Concerning the linear dynamics of \(T\in {{\mathcal {L}}}(X)\), with X a lcHs, the operator T is called supercyclic if, for some \(z\in X\), the projective orbit \(\{\lambda T^nz\,:\ \lambda \in {{\mathbb {C}}},\ n\in {{\mathbb {N}}}_0\}\) is dense in X. Since the closure of the linear span of a projective orbit is separable, if such a supercyclic operator \(T\in {{\mathcal {L}}}(X)\) exists, then X is necessarily separable.
Observe that the space \(H^\infty _v\) is never separable, [24, Theorem 1.1]. Therefore, every operator \(T\in {{\mathcal {L}}}(H^\infty _v)\) is clearly not supercyclic. However, the spaces \(H({{\mathbb {D}}})\), [21, Theorem 27.2.5], and \(H^0_v\), [24, Theorem 1.1], for every weight v are always separable. Hence, the problem of supercyclicity for non-zero operators \(T\in {{\mathcal {L}}}(H({{\mathbb {D}}}))\) and \(T\in {{\mathcal {L}}}(H^0_v)\) arises.
The following result, [5, Theorem 6.4], is stated here for Banach spaces.
Theorem 3.1
Let X be a Banach space and let \(T\in {{\mathcal {L}}}(X)\) be a compact operator such that \(1\in \sigma (T;X)\) with \(\sigma (T;X){\setminus } \{1\}\subseteq \overline{B(0,\delta )}\) for some \(\delta \in (0,1)\) and satisfying \(\textrm{Ker}(I-T)\cap \textrm{Im} (I-T)=\{0\}\). Then T is power bounded and uniformly mean ergodic.
A consequence of the previous theorem is the following result.
Proposition 3.2
Let v be a weight function on [0, 1) satisfying \(\lim _{r\rightarrow 1^-}v(r)=0\). For each \(t\in [0,1)\) both of the operators \(C_t\in {{\mathcal {L}}}(H^\infty _v)\) and \(C_t\in {{\mathcal {L}}}(H^0_v)\) are power bounded, uniformly mean ergodic and fail to be supercyclic.
Proof
Fix \(t\in [0,1)\). It was already noted that \(C_t\in {{\mathcal {L}}}(H^\infty _v)\) cannot be supercyclic. The operator \(C_t\) is a compact operator on both \(H^\infty _v\) and on \(H^0_v\) (cf. Proposition 2.7). Therefore, the compact transpose operators \(C_t'\in {{\mathcal {L}}}((H^\infty _v)')\) and \(C'_t\in {{\mathcal {L}}}((H^0_v)')\) have the same non-zero eigenvalues as \(C_t\) (see, e.g., [15, Theorem 9.10-2(2)]). In view of Proposition 2.8 it follows that \(\sigma _{pt}(C_t';(H^\infty _v)')=\sigma _{pt}(C_t';(H^0_v)')=\{\frac{1}{m+1}\,:\, m\in {{\mathbb {N}}}_0\}\). We can apply [6, Proposition 1.26] to conclude that \(C_t\) is not supercyclic on \(H^0_v\).
By Proposition 2.8 and its proof (as \(x^{[0]}=(t^n)_{n\in {{\mathbb {N}}}_0}\)) we have that \(\textrm{Ker}(I-C_t)=\textrm{span}\{g_0\}\), with \(g_0(z)=\sum _{n=0}^\infty t^nz^n\), for \(z\in {{\mathbb {D}}}\). On the other hand, \(\textrm{Im}(I-C_t)\) is a closed subspace of \(H^\infty _v\) (resp., of \(H^0_v\)), as \(C_t\) is compact in \(H^\infty _v\) (resp., in \(H^0_v\))), and \(\textrm{Im}(I-C_t)\subseteq \{g\in H^\infty _v\,:\, g(0)=0\}\) (resp., \(\subseteq \{g\in H^0_v\,:\, g(0)=0\}\)), because \(C_tf(0)=f(0)\) for each \(f\in H^\infty _v\) (resp., each \(f\in H^0_v\)). Moreover, [15, Theorem 9.10.1] implies that \(\mathrm{codim\,Im}(I-C_t)=\mathrm{dim\,Ker} (I-C_t)=1\). Accordingly, both \(\textrm{Im}(I-C_t)\) and \(\{g\in H^\infty _v\,:\, g(0)=0\}=\textrm{Ker}(\delta _0)\) are hyperplanes, where \(\delta _0\in (H^\infty _v)'\) is the linear evaluation functional \(f\mapsto f(0)\), for \(f\in H^\infty _v\). It follows that necessarily \(\textrm{Im}(I-C_t)=\{g\in H^\infty _v:\, g(0)=0\}\).
Let \(h\in \textrm{Im}(I-C_t)\cap \textrm{Ker}(I-C_t)\). Then \(h(0)=0\) and there exists \(\lambda \in {{\mathbb {C}}}\) such that \(h=\lambda g_0\). This yields that \(0=h(0)=\lambda g_0(0)=\lambda \). Hence, \(h=0\). So, \(\textrm{Im}(I-C_t)\cap \textrm{Ker}(I-C_t)=\{0\}\).
Proposition 2.8 implies that \(1\in \sigma (C_t; H^\infty _v)=\sigma (C_t; H^0_v)=\{\frac{1}{m+1}\,;\, m\in {{\mathbb {N}}}_0\}\cup \{0\}\). Consequently, for \(\delta =\frac{1}{2}\), all the assumptions of Theorem 3.1 are satisfied. So, we can conclude that \(C_t\) is power bounded and uniformly mean ergodic on both \(H^\infty _v\) and on \(H^0_v\). \(\square \)
In contrast to the compactness of \(C_t\) acting in the Banach spaces \(H^\infty _v\) and \(H^0_v\) (cf. Proposition 2.7) the situation for the Fréchet space \(H({{\mathbb {D}}})\) is different.
Proposition 3.3
For each \(t\in [0,1)\) the operator \(C_t:H({{\mathbb {D}}})\rightarrow H({{\mathbb {D}}})\) is an isomorphism and, hence, it is not compact.
Proof
Fix \(t\in [0,1)\). Consider the operator \(T_t:H({{\mathbb {D}}})\rightarrow H({{\mathbb {D}}})\), for \(f\in H({{\mathbb {D}}})\), given by
Then \(T_t\) is clearly well-defined. Moreover, its graph is closed. Indeed, for a given sequence \((f_n)_{n\in {{\mathbb {N}}}}\subset H({{\mathbb {D}}})\), suppose that \(f_n\rightarrow f\) in \(H({{\mathbb {D}}})\) and \(T_tf_n\rightarrow g\) in \(H({{\mathbb {D}}})\). Since multiplication operators (by elements from \(H({{\mathbb {D}}})\)) and the differentiation operator are continuous on \(H({{\mathbb {D}}})\) and the evaluation functionals at points of \({{\mathbb {D}}}\) belong to \(H({{\mathbb {D}}})'\), it follows that \(f'_n\rightarrow f'\) in \(H({{\mathbb {D}}})\) and hence, \(T_tf_n=(1-tz)(f_n+zf'_n)\rightarrow (1-tz)(f+zf')=T_tf\) in \(H({{\mathbb {D}}})\). Accordingly, \(g=T_tf\). Since \(H({{\mathbb {D}}})\) is a Fréchet space, the closed graph theorem, [20, Corollary 5.4.3], implies that \(T_t\in {{\mathcal {L}}}(H({{\mathbb {D}}}))\).
Finally, it is routine to verify that \(C_t\circ T_t=T_t\circ C_t=I\). So, the inverse operator \(C_t^{-1}=T_t\in {{\mathcal {L}}}(H({{\mathbb {D}}}))\) exists and hence, \(C_t\) is a bi-continuous isomorphism of \(H({{\mathbb {D}}})\) onto itself. In particular, \(C_t\) cannot be compact. \(\square \)
Let \(\Lambda :=\{\frac{1}{n+1}:\, n\in {{\mathbb {N}}}_0\}\) and \(\Lambda _0:=\Lambda \cup \{0\}\). We recall from [4, Lemma 2.7] the following lemma, which is an extension of a result of Rhoades [27].
Lemma 3.4
For every \(\mu \in {{\mathbb {C}}}{\setminus } \Lambda _0\) there exist \(\delta =\delta _\mu >0\) and constants \(d_\delta , D_\delta >0\) such that \(\overline{B(\mu ,\delta )}\cap \Lambda _0=\emptyset \) and
where \(\alpha (\nu ):=\textrm{Re}(\frac{1}{\nu })\).
Remark 3.5
As a direct application of Lemma 3.4 we obtain, for every \(\mu \in {{\mathbb {C}}}{\setminus } \Lambda _0\), that there exist \(\delta >0\) and \(d_\delta , D_\delta >0\) such that \(\overline{B(\mu , \delta )}\cap \Lambda _0=\emptyset \) and, for every \(\nu \in B(\mu ,\delta )\) and \(n\in {{\mathbb {N}}}_0\), we have that
for all \(h\in \{1,\ldots , n-1\}\), where \(\alpha (\nu )=\textrm{Re}(\frac{1}{\nu })\).
For each \(k\in {{\mathbb {N}}}\) with \(k\ge 2\) define \(r_k:=(1-\frac{1}{k})\). Define the norms \(\Vert \cdot \Vert _k\) and \(|||\cdot |||_k\) on \(H({{\mathbb {D}}})\) by
and
Lemma 3.6
Each of the sequences \(\{\Vert \cdot \Vert _k\}_{k\ge 2}\) and \(\{|||\cdot |||_k\}_{k\ge 2}\) is a fundamental system of norms for \((H({{\mathbb {D}}}),\tau _c)\).
Proof
Given \(r\in (0,1)\) choose any \(k\ge 2\) such that \(0<r<(1-\frac{1}{k})\). Then, for every \(f\in H({{\mathbb {D}}})\), we have
On the other hand, given \(k\ge 2\), let \(r_k:=(1-\frac{1}{k})<(1-\frac{1}{k+1}):=r_{k+1}\). By the Cauchy inequalities, for \(n\in {{\mathbb {N}}}_0\), we have
and hence,
with \(c=\frac{1}{1-\frac{r_k}{r_{k+1}}}=k^2>0\) as \(\frac{r_k}{r_{k+1}}<1\), which is independent of f.
So, the systems \(\{q_r\}_{r\in (0,1)}\) and \(\{\Vert \cdot \Vert _k\}_{k\ge 2}\) are equivalent on \(H({{\mathbb {D}}})\).
Observe, for every \(k\ge 2\), that
and that
for \(f\in H({{\mathbb {D}}})\), where \(\sum _{n=0}^\infty \left( \frac{r_k}{r_{k+1}}\right) ^n=k^2\). Therefore, the systems \(\{\Vert \cdot \Vert _k\}_{k\ge 2}\) and \(\{|||\cdot |||_k\}_{k\ge 2}\) are equivalent. \(\square \)
Proposition 3.7
For each \(t\in [0,1)\) the spectra of the operator \(C_t\in {{\mathcal {L}}}(H({{\mathbb {D}}}))\) are given by
and
Proof
Let \(t\in [0,1)\) be fixed. For any weight function v on [0, 1) satisfying \(\lim _{r\rightarrow 1^-}v(r)=0\), we have \(H^\infty _v\subseteq H({{\mathbb {D}}})\) continuously and \(\Phi :H({{\mathbb {D}}})\rightarrow \omega \) is a continuous imbedding. Accordingly, \(\sigma _{pt}(C_t;H_v^\infty )\subseteq \sigma _{pt}(C_t;H({{\mathbb {D}}}))\subseteq \Lambda \); see the proof of Proposition 2.8. Since \(\sigma _{pt}(C_t;H^\infty _v)=\Lambda \) (cf. Proposition 2.8) and \(\sigma _{pt}(C_t^\omega ;\omega )=\Lambda \) [5, Theorem 3.7], it follows that \(\sigma _{pt}(C_t;H({{\mathbb {D}}}))=\Lambda \). Moreover, in view of Proposition 2.8 above and Theorem 3.7 in [5], the eigenspace corresponding to each eigenvalue \(\frac{1}{n+1}\in \Lambda \) is 1-dimensional. By Proposition 3.3, the operator \(C_t:H({{\mathbb {D}}})\rightarrow H({{\mathbb {D}}})\) is a bi-continuous isomorphism and so \(0\not \in \sigma (C_t; H({{\mathbb {D}}}))\).
The claim is that \({{\mathbb {C}}}{\setminus } \Lambda _0\subseteq \rho (C_t;H({{\mathbb {D}}}))\). To establish this claim, fix \(\nu \in {{\mathbb {C}}}{\setminus }\Lambda _0\). Given \(g(z)=\sum _{n=0}^\infty c_nz^n\in H({{\mathbb {D}}})\), consider the identity
where \(f(z)=\sum _{n=0}^\infty a_n z^n\in H({{\mathbb {D}}})\) is to be determined. It follows from (1.6) that \(C_tf(z)=\sum _{n=0}^\infty (\frac{t^na_0+t^{n-1}a_1+\cdots +a_n}{n+1})z^n\) from which the identity \((C_t-\nu I)f(z)=\sum _{n=0}^\infty (\frac{t^na_0+t^{n-1}a_1+\cdots +a_n}{n+1}-\nu a_n)z^n\) is clear. So, (3.5) is satisfied if and only if
that is, if and only if
In view of this we can argue, as in the proof of [5, Lemma 3.6], to show that if a function \(f\in H({{\mathbb {D}}})\) exists which satisfies the identity (3.5), then the Taylor coefficients \((a_n)_{n\in {{\mathbb {N}}}_0}\) of f must verify the following equalities
Observe, for each \(n\ge 1\) and \(h\in \{1,\ldots ,n\}\), that
and so
Accordingly, to verify the claim we need to prove that the power series \(\sum _{n=0}^\infty a_nz^n\) is convergent in \({{\mathbb {D}}}\), with \((a_n)_{n\in {{\mathbb {N}}}_0}\) defined according to (3.6). First, observe that the series \(g(z)=\sum _{n=0}^\infty c_nz^n\) is convergent in \({{\mathbb {D}}}\) and satisfies
Therefore, the series \(\sum _{n=1}^\infty A_nz^n\) has the same radius of convergence as the series \(\sum _{n=0}^\infty c_nz^n\) and hence, it converges in \(H({{\mathbb {D}}})\). Accordingly, \(f_1(z):=\sum _{n=1}^\infty A_nz^n\), for \(z\in {{\mathbb {D}}}\), belongs to \(H({{\mathbb {D}}})\). On the other hand, the series
To establish the convergence of the series \(\sum _{n=1}^\infty B_nz^n\) in \(H({{\mathbb {D}}})\), fix \(z\in {{\mathbb {D}}}{\setminus }\{0\}\) and \(r\in (|z|,1)\). Recall, for every \(n\in {{\mathbb {N}}}_0\), that the Taylor coefficients of g satisfy (as \(\frac{1}{r}>1\))
where \(C:=\max _{|\xi |=r}|g(\xi )|\). Therefore, setting \(\alpha :=\alpha (\nu )=\textrm{Re}(\frac{1}{\nu })\) and \(d:=d_\delta \) and \(D:=D_\delta \) for a suitable \(\delta >0\) (cf. Remark 3.5), we obtain via (3.1) and (3.2) that
which is finite after observing that if \(\alpha \le 0\), then \(\left( \frac{n+1}{n-h}\right) ^\alpha =\left( \frac{n-h}{n+1}\right) ^{-\alpha }\le 1\) for every \(h\in {{\mathbb {N}}}\) and every \(n\ge h+1\), whereas if \(\alpha >0\), then \((\frac{n+1}{n-h})^\alpha =(1+\frac{h+1}{n-h})^\alpha \le (2+h)^\alpha \). This implies that the series \(\sum _{n=1}^\infty B^nz^n\) converges in \(H({{\mathbb {D}}})\). Accordingly, \(f_2(z):=\sum _{n=1}^\infty B_nz^n\), for \(z\in {{\mathbb {D}}}\), belongs to \(H({{\mathbb {D}}})\).
Set \(f(z):=\frac{c_0}{1-\nu }+f_1(z)+f_2(z)\), for \(z\in {{\mathbb {D}}}\). Then \(f\in H({{\mathbb {D}}})\). Moreover, the arguments above imply that f satisfies (3.5). The identities (3.6) imply that f is the unique solution of (3.5). Accordingly, the inverse operator \((C_t-\nu I)^{-1}:H({{\mathbb {D}}})\rightarrow H({{\mathbb {D}}})\) exists. In particular, \((C_t-\nu I)^{-1}\in {{\mathcal {L}}}(H({{\mathbb {D}}}))\) as it is the inverse of a continuous linear operator on a Fréchet space.
Since \(\nu \in {{\mathbb {C}}}{\setminus }\Lambda _0\) is arbitrary and \(0\in \rho (C_t;H({{\mathbb {D}}}))\), we can conclude that \(\sigma (C_t;H({{\mathbb {D}}}))=\Lambda \).
It remains to show that \(\sigma ^*(C_t;H({{\mathbb {D}}}))=\Lambda _0\). To establish this, fix \(\mu \in {{\mathbb {C}}}{\setminus }\Lambda _0\) and observe, by Lemma 3.4, that there exist \(\delta >0\) and constants \(d_\delta , D_\delta >0\) such that \(\overline{B(\mu ,\delta )}\cap \Lambda _0=\emptyset \) and the inequalities (3.1) and (3.2) are satisfied. We will show that \(B(\mu ,\delta )\subset \rho (C_t;H({{\mathbb {D}}}))\) and that the set \(\{(C_t-\nu I)^{-1}:\, \nu \in B(\mu ,\delta )\}\) is equicontinuous in \({{\mathcal {L}}}(H({{\mathbb {D}}}))\). To see this, first observe that the function \(\nu \in \overline{B(\mu ,\delta )} \mapsto \textrm{Re}(\frac{1}{\nu })\in {{\mathbb {R}}}\) is continuous and hence, \(\alpha _0:=\max _{\nu \in \overline{B(\mu ,\delta )} }\{\textrm{Re}(\frac{1}{\nu })\}\) exists. For the sake of simplicity of notation set \(d:=d_\delta \) and \(D:=D_\delta \).
Let \(\nu \in B(\mu ,r)\), where \(r:=\frac{1}{2}d(\Lambda _0, \overline{B(\mu ,\delta )})>0\) has the property that \(|\nu -\frac{1}{j}|>r\) for all \(j\in {{\mathbb {N}}}\). It was proved above, for any fixed \(g(z)=\sum _{n=0}^\infty c_nz^n\in H({{\mathbb {D}}})\), that
for each \(z\in {{\mathbb {D}}}\). So, for \(k\ge 2\) fixed, consider the norm \(\Vert \cdot \Vert _k\) in \(H({{\mathbb {D}}})\). Then we have, via (3.6), that
Moreover, (3.1) and (3.2) with \(\alpha (\nu )=\textrm{Re}(\frac{1}{\nu })\le \alpha _0\) imply, for each \(h\in {{\mathbb {N}}}\), that
with \(K:=\max \{d^{-1},d^{-1}D\}\), and hence, since \(|\nu |>r\) for all \(\nu \in B(\mu ,\delta )\), that
with \(K'=\frac{K}{r^2}\sum _{h=1}^\infty t^h\left( 1-\frac{1}{k}\right) ^h(2+h)^{\alpha _0}<\infty \), by the ratio test, for instance.
We have established, for every \(\nu \in B(\mu ,\delta )\), that
Since \(g\in H({{\mathbb {D}}})\) and \(k\ge 2\) are arbitrary, this shows that the set \(\{(C_t-\nu I)^{-1}:\, \nu \in B(\mu ,\delta )\}\) is equicontinuous. Hence, \(\sigma ^*(C_t;H({{\mathbb {D}}}))=\Lambda _0\).\(\square \)
Proposition 3.8
For each \(t\in [0,1)\) the operator \(C_t:H({{\mathbb {D}}})\rightarrow H({{\mathbb {D}}})\) is power bounded, uniformly mean ergodic but, it fails to be supercyclic. Moreover, \( (I-C_t)(H({{\mathbb {D}}}))\) is the closed subspace of \(H({{\mathbb {D}}})\) given by
and we have the decomposition
Proof
Fix \(t\in [0,1)\). We first prove that \(C_t\) is power bounded. Once this is established, \(C_t\) is necessarily uniform mean ergodic because \(H({{\mathbb {D}}})\) is a Fréchet- Montel space (see [1, Proposition 2.8]).
Given \(k\ge 2\) we have, for every \(f\in H({{\mathbb {D}}})\) and with \(r_k:=(1-\frac{1}{k})\), that
because \(r_k^n\le r_k^j\) for all \(j\in \{0,1,\ldots ,n\}\). It follows, for every \(n\in {{\mathbb {N}}}\), that
Since \(k\ge 2\) is arbitrary, the operator \(C_t\in {{\mathcal {L}}}(H({{\mathbb {D}}}))\) is indeed power bounded.
To establish that \(C_t:H({{\mathbb {D}}})\rightarrow H({{\mathbb {D}}})\) is not supercyclic, note that the continuous embedding \(\Phi :H({{\mathbb {D}}})\rightarrow \omega \) has dense range. The operator \(C_t^\omega \in {{\mathcal {L}}}(\omega )\) satisfies \(\Phi \circ C_t=C_t^\omega \circ \Phi \) as an identity in \({{\mathcal {L}}}(H({{\mathbb {D}}}),\omega )\), which implies if \(C_t:H({{\mathbb {D}}})\rightarrow H({{\mathbb {D}}})\) is supercyclic, then also \(C_t^\omega :\omega \rightarrow \omega \) must be supercyclic as \(\Phi \circ C_t^n=\Phi \circ C_t\circ C_t^{n-1}=C_t^\omega \circ \Phi \circ C_t^{n-1}=\cdots =(C_t^\omega )^n\circ \Phi \), for all \(n\in {{\mathbb {N}}}\), and \(\Phi (H({{\mathbb {D}}}))\) is dense in \(\omega \). A contradition with [5, Theorem 6.1].
To establish (3.7) note that \((I-C_t)(H({{\mathbb {D}}}))\subseteq \{g\in H({{\mathbb {D}}})\,:\ g(0)=0\}\) because \(C_tf(0)=f(0)\) for every \(f\in H({{\mathbb {D}}})\). To show the reverse inclusion, let \(g\in H({{\mathbb {D}}})\) satisfy \(g(0)=0\). Then \(h(z):=zg'(z)+g(z)\), for \(z\in {{\mathbb {D}}}\), is holomorphic and \(h(0)=0\). Accordingly, also \(z\mapsto \frac{h(z)}{z}\), for \(z\in {{\mathbb {D}}}{\setminus }\{0\}\), and taking the value \(h'(0)\) at \(z=0\) is holomorphic in \({{\mathbb {D}}}\). Define \(f\in H({{\mathbb {D}}})\) by
and note that \(f(0)=0\). Direct calculation reveals that
from which it follows that
Since \(f(0)=0\), we can conclude that
that is, \((C_t-I)f=g\) and so \(g\in (I-C_t)(H({{\mathbb {D}}}))\). Hence, (3.7) is valid.
To show the validity of (3.8) it suffices to repeat the argument given in the proof of Proposition 3.2. \(\square \)
References
Albanese, A.A., Bonet, J., Ricker, W.J.: Mean ergodic operators in Fréchet spaces. Ann. Acad. Sci. Fenn. Math. 34, 401–436 (2009)
Albanese, A.A., Bonet, J., Ricker, W.J.: The Cesàro operator in growth Banach spaces of analytic functions. Integr. Equ. Oper. Theory 86, 97–112 (2016)
Albanese, A.A., Bonet, J., Ricker, W.J.: The Cesàro operator on Korenblum type spaces of analytic functions. Collect. Math. 69, 263–281 (2018)
Albanese, A.A., Bonet, J., Ricker, W.J.: The Cesàro operator on duals of power series spaces of infinite type. J. Oper. Theory 79, 373–402 (2018)
Albanese, A.A., Bonet, J., Ricker, W.J.: Spectral properties of generalized Cesáro operator in sequence spaces. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 117, Article number 140 (2023). https://doi.org/10.48550/arXiv.2305.04805
Bayart, F., Matheron, E.: Dynamics of Linear Operators. Cambridge Tracts in Mathematics, vol. 179. Cambridge University Press, Cambridge (2009)
Bierstedt, K.D., Bonet, J., Galbis, A.: Weighted spaces of holomorphic functions on balanced domains. Mich. Math. J. 40, 271–297 (1993)
Bierstedt, K.D., Summers, W.H.: Biduals of weighted Banach spaces of analytic functions. J. Aust. Math. Soc. 54, 70–79 (1993)
Blasco, O.: Cesáro-type operators on Hardy spaces. J. Math. Anal. Appl. 529, Article number 127017 (2024). https://doi.org/10.1016/j.jmaa.2023.127017
Bonet, J.: Weighted Banach spaces of analytic functions with sup-norms and operators between them: a survey. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 116(4), Article number 184 (2022)
Bonet, J., Ricker, W.J.: Mean ergodicity of multiplication operators in weighted spaces of holomorphic functions. Arch. Math. 92, 428–437 (2009)
Curbera, G.P., Ricker, W.J.: The Cesàro operator and unconditional Taylor series in Hardy spaces. Integr. Equ. Oper. Theory 83, 179–195 (2015)
Curbera, G.P., Ricker, W.J.: Fine spectra and compactness of generalized Cesàro operators in Banach lattices in \({\mathbb{C} }^{\mathbb{N} }_{0}\). J. Math. Anal. Appl. 507, 125824 (2022)
Danikas, N., Siskakis, A.: The Cesàro operator on bounded analytic functions. Analysis 13, 295–299 (1993)
Edwards, R.E.: Functional Analysis, Theory and Applications. Holt, Rinehart and Winston, New York (1965)
Galanopoulos, P., Girela, D., Merchán, N.: Cesáro-like operators acting on spaces of analytic functions. Anal. Math. Phys. 12, Article number 51 (2022)
Grosse-Erdmann, K.-G., Peris Manguillot, A.: Linear Chaos, Universitext. Springer, London (2011)
Grothendieck, A.: Topological Vector Spaces. Gordon and Breach, London (1973)
Hardy, G.H., Littlewood, J.E., Pólya, G.: Inequalities. Cambridge University Press, Cambridge (1934)
Jarchow, H.: Locally Convex Spaces. Teubner, Stuttgart (1981)
Köthe, G.: Topological Vector Spaces I, 2nd edn. Springer, Berlin (1983)
Köthe, G.: Topological Vector Spaces II. Springer, Berlin (1979)
Krengel, U.: Ergodic Theorems. Walter de Gruyter, Berlin (1985)
Lusky, W.: On the isomorphism classes of weighted spaces of harmonic and holomorphic functions. Studia Math. 175, 19–40 (2006)
Rhaly, H.C., Jr.: Discrete generalized Cesàro operators. Proc. Am. Math. Soc. 86, 405–409 (1982)
Rhaly, H.C., Jr.: Generalized Cesàro matrices. Canad. Math. Bull. 27, 417–422 (1984)
Rhoades, B.E.: Spectra of some Hausdorff operators. Acta Sci. Math. (Szeged) 32, 91–100 (1971)
Sawano, Y., El-Shabrawy, S.R.: Fine spectra of the discrete generalized Cesàro operator on Banach sequence spaces. Monatsh. Math. 192, 185–224 (2020)
Waelbrock, L.: Topological Vector Spaces and Algebras, LNM, vol. 230. Springer, Berlin (1971)
Yildrim, M., Durna, N.: The spectrum and some subdivisions of the spectrum of discrete generalized Cesàro operators on \(\ell ^p\) (\(1<p<\infty \)). J. Inequal. Appl. 2017(1), 1–13 (2017)
Yildrim, M., Mursaleen, M., Dogan, C.: The spectrum and fine spectrum of the generalized Rhaly–Cesàro matrices on \(c_0\) and \(c\). Oper. Matrices 12(4), 955–975 (2018)
Acknowledgements
The research of J. Bonet was partially supported by the Project PID2020-119457GB-100 funded by MCIN/AEI/10.13039/501100011033 and by “ERFD A way of making Europe” and by the project GV AICO/2021/170.
Funding
Open access funding provided by Università del Salento within the CRUI-CARE Agreement.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Albanese, A.A., Bonet, J. & Ricker, W.J. Generalized Cesàro operators in weighted Banach spaces of analytic functions with sup-norms. Collect. Math. (2024). https://doi.org/10.1007/s13348-024-00437-9
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s13348-024-00437-9
Keywords
- Generalized Cesáro operator
- Weighted Banach spaces of analytic functions
- Compact operator
- Spectrum
- Supercyclic
- Mean ergodic
- Power bounded