Semigroups of weighted composition operators on spaces of holomorphic functions

We recall some of the standard facts about one-parameter semigroups of operators, which may be found in many places, such as [29] and [43].

A uniformly continuous semigroup $(T_{t})_{t\geq 0}$ is one satisfying $\|T_{t}-{\rm Id}\|\to 0$ as $t\to 0^{+}$.

Associated with this is the notion of an infinitesimal generator, or simply generator. We define an (in general unbounded) operator $A$ whose domain is

and then

Moreover, the generator of a $C_{0}$-semigroup characterizes completely a semigroup, that is two $C_{0}$-semigroups are equal if and only if their generators are equal.

By the uniform boundedness principle, each $C_{0}$-semigroup is uniformly bounded on each compact interval. As a corollary, for every $C_{0}$-semigroup $(T_{t})_{t\geq 0}$, there exists $w\in\mathbb{R}$ and $M\geq 1$ such that

Contractive $C_{0}$-semigroups are the ones for which one can take $M=1$ and $w=0$, whereas quasicontractive $C_{0}$-semigroups are the ones for which one can take $M=1$ and $w$ is arbitrary.

The domain $D(A)$ of the generator $A$ of a $C_{0}$-semigroup is always dense in $X$ (and moreover $(A,D(A))$ is a closed operator). It is then natural to characterize the linear operators $(A,D(A))$ that are the generator of $C_{0}$-semigroups. For semigroups of contractions on Hilbert spaces, Lumer and Phillips, in 1961, provided a beautiful criterion [38] (see also [29, Theorem 3.15]).

Since $(T_{t})_{t\geq 0}$ is a $C_{0}$-semigroup of quasicontractions on a Hilbert space $H$ if and only if there exists $w\geq 0$ such that $(e^{-wt}T_{t})_{t\geq 0}$ is a $C_{0}$-semigroup of contractions, it is then easy to deduce a version of the Lumer–Phillips Theorem for $C_{0}$-quasicontractions.

For generators of $C_{0}$-semigroups which are not necessarily quasicontractive, another beautiful criterion involving the growth of the resolvent is due to Hille and Yosida [29, Theorem 3.8].

We shall restrict ourselves to Banach spaces $X$ of functions that are holomorphic on a domain $\Omega$ (usually the unit disc $\mathbb{D}$ but sometimes the right half-plane $\mathbb{C}_{+}$ or the complex plane $\mathbb{C}$) and satisfying the condition that point evaluations $f\mapsto f(z)$ are continuous for all $z\in\Omega$. Assuming this, we have that norm convergence of a sequence $(f_{n})$ to $f$ implies local uniform convergence (uniform convergence on compact subsets of $\Omega$).

Recall that for suitable $\phi:\Omega\to\Omega$ holomorphic, the composition operator $C_{\phi}:X\to X$ is defined by $(C_{\phi}f)(z)=f(\phi(z))$, for $f\in X$ and $z\in\Omega$ (assuming that $C_{\phi}$ maps $X$ boundedly into $X$, an issue which will be discussed later). Likewise, for suitable $w$ holomorphic on $\Omega$, the weighted composition operator $W_{w,\phi}$ on $X$ is defined by $(W_{w,\phi}f)(z)=w(z)f(\phi(z))$.

The main theme of this paper is to study $C_{0}$-semigroups of (weighted) composition operators. However, we may also look at composition semigroups from a non-operatorial point of view (for example, as in [16]).

In this situation there exists a unique holomorphic function $G:\Omega\to\mathbb{C}$ such that

This function is called the infinitesimal generator of the analytic semiflow $(\phi_{t})_{t\geq 0}$ on $\Omega$ and we denote by $\mathop{\rm Gen}\nolimits(\Omega)$ the set of all infinitesimal generator of analytic semiflows on $\Omega$.

There are several complete characterization of $\mathop{\rm Gen}\nolimits(\mathbb{D})$ in [16, Chap. 10], which is discussed in more details in Section 4.2.

Analytic semiflows on $\mathbb{D}$ can be partitioned into two classes, depending on the localization of their Denjoy–Wolff point $\alpha$, discussed below in Section 4 (see [1], [24, Chap. 2] and [16, Chap. 8]).

If $\alpha\in\mathbb{D}$, by conjugating by the automorphism $b_{\alpha}$, where

we may suppose without loss of generality that $\alpha=0$. In this case there is a semiflow model

where $c\in\mathbb{C}$ with $\mathop{\rm Re}\nolimits c\geq 0$, and $h:\mathbb{D}\to\Omega$ is a conformal bijection between $\mathbb{D}$ and a domain $\Omega\subset\mathbb{C}$, with $h(0)=0$ and $\Omega$ is spiral-like or star-like (if $c$ is real), in the sense that

If $\alpha\in\mathbb{T}$, then there exists a conformal map $h$ from $\mathbb{D}$ onto a domain $\Omega$ such that $\Omega+it\subset\Omega$ for all $t\geq 0$, and there is a semiflow model

This subsection relies heavily on Subsection 8.2 in [16].

If $\phi_{t}$ is an automorphism for all $t\geq 0$, we can introduce the notation $\phi_{-t}:=\phi_{t}^{-1}$ for all $t\geq 0$ and then observe that

• c)

  $\phi_{s+t}=\phi_{s}\circ\phi_{t}$ for all $s,t\in\mathbb{R}$;

• d)

  for all $z\in\mathbb{D}$, the mapping $t\mapsto\phi_{t}(z)$ is continuous on $\mathbb{R}$ if $(\phi_{t})_{t\geq 0}$ is a continuous flow.

In fact, d) is equivalent to the continuity of $t\mapsto\phi_{t}\in\mathop{\rm Hol}\nolimits(\mathbb{D})$ on $\mathbb{R}$, where $\mathop{\rm Hol}\nolimits(\mathbb{D})$ is endowed with the topology of the uniform convergence on compacta of $\mathbb{D}$.

Here is a characterization of continuous flows in the set of continuous semiflows [16, Thm. 8.2.4].

The following theorem [16, Thm. 8.2.6] is an explicit description of all the continuous flows on $\mathbb{D}$.

Clearly a semiflow $(\phi_{t})_{t\geq 0}$ induces a semigroup of composition operators (on $\mathbb{D}$ these are bounded, by Littlewood’s subordination theorem [37]), and the following condition gives a way of testing the strong continuity.

In particular, if the polynomials are dense in $X$ then it is enough to check that $\lim_{t\to 0}\|T_{t}e_{n}-e_{n}\|_{X}=0$ for all $n=0,1,2,\ldots$, where $e_{n}(z)=z^{n}$.

In Proposition 2.10 we have seen a sufficient condition for a semiflow to induce a $C_{0}$-semigroup of composition operators, and in the Hardy, Dirichlet and Bergman spaces we do indeed arrive at such a semigroup.

Recently, Gallardo-Gutiérrez, Siskakis and Yakubovich [32] have shown that weighted composition operators $(W_{w_{t},\phi_{t}})_{t\geq 0}$ do not form a nontrivial $C_{0}$-semigroup on spaces $X$ satisfying $H^{\infty}\subset X\subset\mathcal{B}$, where $\mathcal{B}$ is the Bloch space. This includes the case $X=\rm BMOA$. The proof is based on estimates for derivatives of interpolating Blaschke products.

For composition operators, and with $X=H^{\infty}$ and $\mathcal{B}$, this result was shown earlier by Blasco et al [15] with an argument involving the Dunford–Pettis property. For spaces between $H^{\infty}$ and $\mathcal{B}$, the result for composition operators was given by Anderson, Jovovic and Smith [2] using geometric function theory.

Rota [45, 46] introduced the concept of a universal operator.

That is, a universal operator is a “model” for all $T\in\mathcal{L}(H)$. Universal operators are of interest in the study of the invariant subspace problem, whether every operator on a separable infinite-dimensional Hilbert space has a nontrivial closed invariant subspace. This has a positive solution if and only every minimal invariant subspace of a given universal operator is finite-dimensional. We refer to [21] for more details and examples.

Caradus [17] gives a convenient sufficient condition for a Hilbert space operator to be universal.

Consider now the hyperbolic automorphisms

with $r\in(-1,1)$. It is helpful to consider them using the parametrization

for $t\in\mathbb{R}$.

It was shown by Nordgren, Rosenthal and Wintrobe [42] that for $r\neq 0$ the operator $C_{\phi_{r}}-{\rm Id}$ is universal on $H^{2}$. Of course $C_{\phi_{r}}$ and $C_{\phi_{r}}-{\rm Id}$ have the same invariant subspaces.

A simpler proof using the fact that $C_{\phi_{r}}$ can be embedded in a $C_{0}$-group was given by Cowen and Gallardo-Gutiérrez [25].

Some work on the invariant subspaces of such operators is due to Matache [39], Mortini [41] and Gallardo-Gutiérrez and Gorkin [31].

Another application of weighted composition operators goes back to Banach [11], who showed that every surjective isometry $F$ of the space $C(K)$ of continuous complex functions on a compact metric space $K$ has the form

where $w\in C(K)$ satisfies $|w|=1$ and $\phi$ is a homeomorphism of $K$.

The Hardy space $H^{2}(\mathbb{D})$ is a Hilbert space, and thus has many (linear) isometries; however for other $p$ with $1<p<\infty$ there are relatively few, and they are expressible as weighted composition operators. In [27] deLeeuw, Rudin, and Wermer gave a description of the isometric surjections of $H^{1}(\mathbb{D})$, which arise from conformal mappings of $\mathbb{D}$ onto $\mathbb{D}$.

Moreover, Forelli [30] gave the following theorem, which does not assume surjectivity.

It is well known that composition operators on the Hardy space $H^{2}(\mathbb{C}_{+})$ of the right half-plane are unitarily equivalent to weighted composition operators on $H^{2}(\mathbb{D})$. For example the following explicit formula is given in [20].

So for example the $C_{0}$-group $(T_{t})_{t\in\mathbb{R}}$ on $H^{2}(\mathbb{C}_{+})$ given by $T_{t}g(z)=g(e^{t}z)$ $(z\in\mathbb{C}_{+})$ for $g\in H^{2}(\mathbb{C}_{+})$ is unitarily equivalent to the weighted composition group $(S_{t})_{t\in\mathbb{R}}$ given by

Formulae for general domains are given, for example, in [35, Prop. 2.1]. If $W$ is a weighted composition operator between two Hardy–Smirnoff spaces $E^{2}(\Omega_{1})$ and $E^{2}(\Omega_{2})$, with $\Omega_{1}$ and $\Omega_{2}$ conformally equivalent to the disc $\mathbb{D}$, then $W$ is unitarily equivalent to a weighted composition operator on $H^{2}(\mathbb{D})$. Similar formulae are given for Bergman spaces.

For a fixed composition operator $C_{\phi}$ there are several possible modes of convergence for the sequence $(C^{n}_{\phi})_{n\geq 1}$, some of which we now list in progressively weaker order.

• •

  Norm convergence. There exists an operator $P\in\mathcal{L}(X)$ such that $\|C_{\phi}^{n}-P\|\to 0$.

• •

  Strong convergence. There exists an operator $P\in\mathcal{L}(X)$ such that $\|C_{\phi}^{n}x-Px\|\to 0$ for all $x\in X$.

• •

  Weak convergence. There exists an operator $P\in\mathcal{L}(X)$ such that $C_{\phi}^{n}x\to Px$ weakly for all $x\in X$.

In each case $P$ is the projection onto $\mathop{\rm fix}\nolimits(C_{\phi}):=\{x\in X:C_{\phi}x=x\}$ along the decomposition

The following theorem from [7] helps with the analysis.

In this case $P$ is the residue at 1.

We sketch the proof.

Recall that $X\hookrightarrow\mathop{\rm Hol}\nolimits(\mathbb{D})$ means that $X$ embeds continuously in $\mathop{\rm Hol}\nolimits(\mathbb{D})$, which means that, for all $\lambda\in\mathbb{D}$, $\delta_{\lambda}:f\mapsto f(\lambda)$ from $X$ to $\mathbb{C}$ is bounded.

Arendt and Batty [3] have given criteria for strong convergence. More recently, from [5] we mention the following theorem.

So it remains to study the essential spectral radius of $C_{\phi}$. To this end we write $\phi^{(n)}=\underbrace{\phi\circ\ldots\phi}_{\hbox{ $n$ factors}}$.

We now address the question whether we can deduce the asymptotic behaviour of a $C_{0}$ semigroup $(C_{\phi_{t}})_{t\geq 0}$ from properties of its generator.

As we shall explain in more detail below, for $X\hookrightarrow\mathop{\rm Hol}\nolimits(\mathbb{D})$ with $(C_{\phi_{t}})_{t\geq 0}$ a $C_{0}$-semigroup induced by an analytic semiflow $(\phi_{t})_{t\geq 0}$ with

the generator $A$ is given by $Af=Gf^{\prime}$ ($f\in D(A)$) with dense domain $D(A)=\{f\in X:Gf^{\prime}\in X\}$.

Various well-known properties of analytic semiflows are the following, which can be found in the recent book [16].

1. 1.

   For all $t\geq 0$, $\phi_{t}$ is injective.

2. 2.

   If there is a $t_{0}>0$ such that $\phi_{t_{0}}$ is an (elliptic) automorphism, then $\phi_{t}$ is an elliptic automorphism for all $t>0$.

3. 3.

   For all semiflows that are not elliptic automorphisms, there is a unique $\alpha\in\overline{\mathbb{D}}$ such that $\lim_{t\to 0}\|\phi_{t}-\alpha\|_{\infty,K}=0$ for all compact subsets $K\subset\mathbb{D}$. Such an $\alpha$ is called the Denjoy–Wolff point of $(\phi_{t})_{t\geq 0}$ (see also Section 4.1).

The following classical theorem of Berkson and Porta [13] describes a semigroup in terms of its generator.

Another characterization was given in [9, Thm. 3.9]. If one knows a priori that $G\in H^{1}(\mathbb{D})$, the necessary and sufficient condition is that $\mathop{\rm Re}\nolimits(\overline{z}G^{*}(z))\leq 0$ a.e. on $\mathbb{T}$, where $G^{*}$ denotes the radial limit of $G$. The proof relies on the following observation which can be seen as a maximum principle for harmonic functions. If $h:\mathbb{D}\to\mathbb{C}$ is in $H^{1}(\mathbb{D})$, then $h(z)=\frac{1}{2\pi}\int_{0}^{2\pi}\mathop{\rm Re}\nolimits\left(\frac{e^{it}+z}{e^{it}-z}\right)f^{*}(e^{it})dt$, where $h^{*}$ stand for the radial limit of $h$, and thefore if $\mathop{\rm Re}\nolimits(h^{*}(e^{it}))\leq 0$ a.e. on $\mathbb{T}$, then $\mathop{\rm Re}\nolimits(h(z))\leq 0$ for all $z\in\mathbb{D}$.

We have already mentioned that $F$ and thus $G$ are always in $H^{p}(\mathbb{D})$ for $0<p<1$, which implies the existence of $G^{*}$ in $L^{p}(\mathbb{T})$. Nevertheless it is not possible to improve [9, Thm. 3.9] since $G(z):=-z\left(\frac{z-1}{z+1}\right)$ is in $H^{p}(\mathbb{D})$ for all $p<1$, satisfies $\mathop{\rm Re}\nolimits(\overline{z}G^{*}(z))\leq 0$ a.e. on $\mathbb{T}$ and is not the generator of a semiflow using (2).

Combining [16, Thm. 10.2.6] which is called “Abate’s formula” and [16, Thm. 10.2.10], we obtain the following characterization of generators.