1 Introduction

The theory of semigroups of weighted composition operators on Hardy and Dirichlet spaces of the unit disc \({{\mathbb {D}}}\) is well understood (see [1, 3–6, 12, 15, 16]), and these have generators that are first order differential operators of the form \(A: f \mapsto G_0 f + G_1 f'\).

Less attention has been given to higher order generators (some of which are the counterpart of those appearing in the theory of heat semigroups), and that is the main theme of this paper.

In Sect. 2 we use the theory of forms to give sufficient conditions for the existence of semigroups on the Hardy space \(H^2\) with generators given by second-order differential operators; other techniques involved in this section include perturbation theory and the use of generalized heat kernels in general reproducing kernel Hilbert spaces.

Then in Sect. 3 we consider a particular class of weighted Hardy spaces, including the Dirichlet space, and by means of numerical range techniques and the Lumer–Phillips theorem, together with explicit expressions for the reproducing kernels, we are able to provide necessary and sufficient conditions for generators of the form \(f \mapsto G f^{(n_0)}\) to generate a semigroup of quasicontractions.

Take \((\beta _n)_{n \ge 0}\) a sequence of positive real numbers. Then \(H^2(\beta )\) is the space of analytic functions

$$\begin{aligned} f(z)=\sum _{n=0}^\infty c_n z^n \end{aligned}$$

in the unit disc that have finite norm

$$\begin{aligned} \Vert f\Vert _\beta = \left( \sum _{n=0}^\infty |c_n|^2 \beta _n^2 \right) ^{1/2}. \end{aligned}$$

The case \(\beta _n=1\) gives the usual Hardy space \(H^2\); also, the case \(\beta _0=1\) and \(\beta _n=\sqrt{n}\) for \(n\ge 1\) provides the Dirichlet space \({\mathcal {D}}\), which is included in \(H^2({{\mathbb {D}}})\).

With an extra condition on \((\beta _n)_n\), the Hilbert space \(H^2(\beta )\) is also a reproducing kernel space, i.e., for all \(w\in {{\mathbb {D}}}\), there exists a function \(k_w\in H^2(\beta )\) such that

$$\begin{aligned} \big \langle f,k_w\big \rangle =f(w), \end{aligned}$$

for all \(f\in H^2(\beta )\) (see p. 19 in [7] and p. 146 in [13]). More precisely, if \((\beta _n)_n\) is such that

$$\begin{aligned} \sum _{n\ge 0}\frac{|w|^{2n}}{\beta _n^2} <\infty \quad \text{ for } \text{ all } \quad w\in {{\mathbb {D}}}, \end{aligned}$$
(1)

it follows that \(H^2(\beta )\) is a reproducing kernel Hilbert space and

$$\begin{aligned} k_w(z)=\sum _{n\ge 0}\frac{{\overline{w}}^n}{\beta _n^2}z^n \qquad \text{ with } \qquad \Vert k_w\Vert ^2_{H^2(\beta )}=\sum _{n\ge 0}\frac{|w|^{2n}}{\beta _n^2}. \end{aligned}$$

In fact (1) is also equivalent to the more explicit condition \(\liminf (\beta _n)^{1/n}\ge 1\).

We recall that \(H^\infty \) is the Banach algebra of bounded analytic functions in the disc, with the supremum norm.

Before embarking on the general theory we give a simple motivating example.

Example 1.1

Let \((Af)(z)=-z^2 f''(z)\) for \(f \in D(A) \subset H^2\), where \(D(A)= \{f \in H^2: f'' \in H^2 \}\). Note that if \(f(z)=z^n\), then \((Af)(z)= -n(n-1) z^n\). It follows easily that A is the generator of the semigroup \((T(t))_{t \ge 0}\), where

$$\begin{aligned} T(t) \sum _{n=0}^\infty a_n z^n = \sum _{n=0}^\infty a_n e^{-n(n-1)t} z^n. \end{aligned}$$

2 Holomorphic semigroups on \(H^2\)

2.1 Application of the theory of forms

First we recall the definition of a form and we present the result that we will apply in order to obtain sufficient conditions for the existence of an analytic semigroup with a prescribed generator involving the second derivative.

The following presentation is based on [2], where we add a simplification concerning the mapping j defined below.

Let V be a complex Hilbert space and \(a:V\times V\rightarrow {{\mathbb {C}}}\) a form, that is a continuous, sesquilinear and coercive mapping, i.e. there exists \(\alpha >0\) such that, for all \(u\in V\),

$$\begin{aligned} \mathrm{Re}\,a(u,u)\ge \alpha \Vert u\Vert ^2_V. \end{aligned}$$
(2)

Then we take H another complex Hilbert space such that there is an embedding \(j:V\rightarrow H\) continuous and dense range. Often we identify j(v) with v for v in V.

For \(x,y\in H\), we say that \(x\in D(A)\) and \(Ax=y\) if \(x\in V\) and

$$\begin{aligned} a(x,v)=\big \langle y,v\big \rangle _H, \end{aligned}$$

for all \(v\in V\).

Then the linear operator A is well-defined, linear, densely-defined, with \(A:D(A)\rightarrow H\).

Moreover, we say that a is j-elliptic if there exists \(w\in {{\mathbb {R}}}\), \(\alpha >0\) such that

$$\begin{aligned} \mathrm{Re}(a(u,u)) +w\Vert u\Vert ^2_H\ge \alpha \Vert u\Vert _V^2. \end{aligned}$$
(3)

The link between a and A is the following.

Theorem 2.1

[2, Thm.4.3] If a is a form which is j-elliptic, then there exists a sector \(\Sigma _\theta \) with \(\theta \in (0,\pi /2]\) such that \(-A\) generates a holomorphic \(C_0\)-semigroup of quasicontractions on H with

$$\begin{aligned} \Vert T(t)\Vert \le e^{\mathrm{Re}(t)w}, \end{aligned}$$

for all t in \(\Sigma _\theta \).

We now present an application of the above theory from forms to semigroups.

Take \(H=H^2\), \(G_1,G_2,G_3\) in \(H^\infty \) such that

$$\begin{aligned} \mathrm{Re}(G_1)\ge \varepsilon _1 \end{aligned}$$
(4)

where \(\varepsilon _1\) is a positive numerical constant.

Then define a by

$$\begin{aligned} a(f,g)=\big \langle G_1f',g'\big \rangle _{H^2}+ \big \langle G_2f,g\big \rangle _{H^2}+ \big \langle G_3f',g\big \rangle _{H^2} \end{aligned}$$

on \(V\times V\), where \(V:=\{f\in H^2 :f'\in H^2\}\) is a Hilbert space endowed with the norm

$$\begin{aligned} \Vert f\Vert _V=\sqrt{ \Vert f\Vert ^2_{H^2} + \Vert f'\Vert ^2_{H^2}}. \end{aligned}$$

Using the Cauchy–Schwarz inequality, a is continuous and obviously sesquilinear. It remains to check that a is j-elliptic.

Note that

$$\begin{aligned} \mathrm{Re}\left( a\left( f,f\right) \right)= & {} \frac{1}{2\pi }\int _0^{2\pi } \mathrm{Re}\left( G_1\left( e^{i\theta }\right) \right) |f'\left( e^{i\theta }\right) |^2d\theta \\&+ \frac{1}{2\pi }\int _0^{2\pi } \mathrm{Re}\left( G_2\left( e^{i\theta }\right) \right) |f\left( e^{i\theta }\right) |^2d\theta + \mathrm{Re}\left( \big \langle G_3 f',f\big \rangle _{H^2}\right) \end{aligned}$$

Moreover, once more using the Cauchy–Schwarz inequality, we have

$$\begin{aligned} \left| \mathrm{Re}\left( \big \langle G_3 f',f\big \rangle _{H^2}\right) \right| \le \Vert G_3\Vert _\infty \Vert f'\Vert _{H^2}\Vert f\Vert _{H^2}\le \frac{\Vert G_3\Vert _\infty }{2}\left( \Vert f'\Vert ^2_{H^2} + \Vert f\Vert ^2_{H^2}\right) . \end{aligned}$$

Now for \(\Vert G_3\Vert _\infty <2\epsilon _1\), and for w sufficiently large, it follows that a is indeed j-elliptic.

The last step consists in identifying A. In our case A is defined by \(Af=g\) when

$$\begin{aligned} \big \langle G_1f',h'\big \rangle _{H^2} +\big \langle G_2 f,h\big \rangle _{H^2} + \big \langle G_3f',h\big \rangle _{H^2} =\big \langle g,h\big \rangle _{H^2}, \end{aligned}$$

for all \(h\in V\).

Note that for all \(k,h\in V\), we have

$$\begin{aligned} \big \langle z(zk)',h\big \rangle _{H^2} =\big \langle k,h'\big \rangle _{H^2}. \end{aligned}$$

Indeed, one can easily check this identity for k and h equal to powers of z.

It follows that A is defined on \(D(A)\subset V\) by

$$\begin{aligned} Af=z\left( zG_1f'\right) ' +G_2f + G_3f'. \end{aligned}$$
(5)

We have proved the following theorem which, as the referee has observed, is formally related to the Black–Scholes equation of mathematical finance (see [8, 9]).

Theorem 2.2

Let \(G_1,G_2,G_3\in H^\infty \) and \(\epsilon _1>0\) be such that

$$\begin{aligned} \mathrm{Re}(G_1)\ge \epsilon _1 \text{ and } \Vert G_3\Vert _\infty <2\epsilon _1. \end{aligned}$$

Let \(A:D(A)\rightarrow H^2\) be defined by

$$\begin{aligned} Af(z)=z(zG_1(z)f'(z))'+G_2(z)f(z)+G_3(z)f'(z). \end{aligned}$$

Then \(-A\) generates a holomorphic \(C_0\)-semigroup of quasicontractions.

An easy example here is given by \(G_1(z)=1\), \(G_2(z)=0\) and \(G_3(z)=-z\), in which case \(Af(z)=z^2f''(z)\), an example discussed in the introduction.

The proof of Theorem 2.2 leads to the following particular cases.

Theorem 2.3

Let \(G_1,G_2,G_3\in H^\infty \) and \(\epsilon _1>0\) be such that

$$\begin{aligned} \mathrm{Re}(G_1)\ge \epsilon _1 \text{ and } \Vert G_3\Vert _\infty <2\epsilon _1. \end{aligned}$$

If moreover there exists \(\varepsilon _2>0\) such that

$$\begin{aligned} \mathrm{Re}(G_2)\ge \epsilon _2 \text{ and } \Vert G_3\Vert _\infty <2\epsilon _2, \end{aligned}$$

then \(-A\) defined by (5) generates a holomorphic \(C_0\)-semigroup of contractions.

Example 2.4

A particular case of Theorem 2.2 is the case where \(Af(z)=z(zG_1(z)f'(z))'\) for which we can conclude that \(-A\) generates a holomorphic \(C_0\)-semigroup of quasicontractions \((T(z))_{z\in \Sigma _\theta }\) with \(\Vert T(z)\Vert \le e^{w\mathrm{Re}(z)}\) for all \(w>0\). This implies that T is a holomorphic \(C_0\)-semigroup of contractions.

2.2 Perturbation theory

Recall that an operator B is bounded relative to another operator A if \(D(A) \subseteq D(B)\) and there are constants \(a,b>0\) such that

$$\begin{aligned} \Vert Bx\Vert \le a\Vert Ax\Vert + b \Vert x\Vert \qquad \hbox {for all} \quad x \in D(A). \end{aligned}$$
(6)

We write \(a_0=\inf \{a>0: \exists b \hbox { such that }\) (6) \(\hbox { holds}\}\), and call this the A-bound of B.

Theorem 2.5

[11, Thm. III.2.10] Suppose that A generates an analytic semigroup \(((T(z))_{z \in \Sigma _\theta \cup \{0\}}\). Then there is an \(\alpha >0\) such that \(A+B\) (with \(D(A+B)=D(A)\)) generates an analytic semigroup for all B with A-bound \(a_0< \alpha \).

More precisely, there is a \(C \ge 1\) such that \(\Vert (\lambda I-A)^{-1}\Vert \le \frac{C}{|\lambda |}\) for \(\lambda \) in some large sector \(\Sigma _{\pi /2+\delta }\) with \(\delta >0\). Then we may take \(\alpha =1/(C+1)\).

To see an application of this result, let us take A defined by \(Af=-z^2 f''\) for \(f \in D(A) \subset H^2\), which is diagonalisable with \(Ae_n=-n(n-1)e_n\), where \(e_n(z)=z^n\) for \(n =0,1,2, \ldots \)

Now

$$\begin{aligned} (\lambda I-A)^{-1}e_n = \frac{e_n}{\lambda +n(n-1)}. \end{aligned}$$

Suppose that \(\lambda =-x+iy\) with \(|x/y| \le \epsilon \) (the case \(x \ge 0\) is easier); then

$$\begin{aligned} \frac{1}{\sqrt{(-x+n(n-1))^2+y^2}} \le \frac{1}{|y|}=\frac{\sqrt{|x/y|^2+1}}{\sqrt{x^2+y^2}} \le \frac{\sqrt{1+\epsilon ^2}}{|\lambda |}, \end{aligned}$$

so we may take \(C=\sqrt{1+\epsilon ^2}\) for any \(\epsilon >0\), and hence \(\alpha =\frac{1}{1+\sqrt{1+\epsilon ^2}}\). Thus, taking \(a_0=1/2\). we may apply Theorem 2.5, and conclude the following.

Theorem 2.6

Let \(Af=-z^2f''\) and \(Bf=gf''\), where \(\Vert g\Vert _\infty \le \frac{1}{2}\). Then \(A+B\) generates a holomorphic semigroup on \(H^2\).

2.3 Heat kernels

It is well known (see, e.g., [10]) that many parabolic partial differential equations on \({{\mathbb {R}}}^n\) can be solved in terms of a (generalized) heat kernel. The simplest example is defined on \(L^2({{\mathbb {R}}})\) by

$$\begin{aligned} T(t)f(s)=(4 \pi t)^{-1/2} \int _{{\mathbb {R}}}f(r) e^{-(s-r)^2/4t} \, dr, \end{aligned}$$

giving a semigroup whose generator is the closure of the Laplacian \(f \mapsto f''\).

Let H be a reproducing kernel Hilbert space of analytic functions on \({{\mathbb {D}}}\). For a semigroup \((T(t))_{t \ge 0}\) on H with generator \(f \mapsto \sum _{k=0}^{n_0} G_k f^{(k)}\), where the \(G_k\) are analytic in \({{\mathbb {D}}}\), define the associated kernel K(t, z, w) on \({{\mathbb {R}}}_+ \times {{\mathbb {D}}}\times {{\mathbb {D}}}\) by

$$\begin{aligned} K(t,z,w)= T(t)k_w(z), \end{aligned}$$

where \(k_w\) is the reproducing kernel for H. That is, K satisfies the equation

$$\begin{aligned} \frac{\partial K}{\partial t}= \sum _{k=0}^{n_0} G_k \frac{\partial ^k K}{\partial z^k}, \qquad K(0,z,w)=k_w(z). \end{aligned}$$
(7)

Proposition 2.7

Suppose that \(T(t)f(w)=F(t,w)\); then

$$\begin{aligned} F(t,w)=\big \langle f, K({t,z,w}) \big \rangle _{z}, \qquad (f \in H). \end{aligned}$$

Proof

This is clearly true if f is a finite linear combination of reproducing kernels, and it follows for all f by the boundedness of the operator T(t). \(\square \)

Theorem 2.8

If the equation \(\sum _{k=0}^{n_0} G_n \varphi ^{(k)}=\lambda \varphi \) has a normalized basis \((\varphi _n)\) of eigenvectors in H, with eigenvalues \((\lambda _n)\), such that \(\mathrm{Re}\,\lambda _n \le 0\) for all n, then

$$\begin{aligned} K(t,z,w)=\sum \varphi _n(z) \overline{\varphi _n(w)} e^{-\lambda _n t}. \end{aligned}$$

Proof

By a simple calculation \(f(w)=\big \langle f,K(0,z,w)\big \rangle _z \) for \(f \in H\), and so \(K(0,z,w)=k_w(z)\). Moreover, the hypotheses on \((\varphi _n)\) and \(\lambda _n\) easily imply that (7) holds. \(\square \)

Example 2.9

For \(H=H^2\), with reproducing kernel \(k_w(z)=1/(1-{{\overline{w}}} z)\), consider the generator \(f \mapsto Gf''\), where \(G(z)=-z^2\), and \(\varphi _n(z)=z^n\), \(\lambda _n=-n(n-1)\). Then

$$\begin{aligned} K(t,z,w)= \sum _{n=0}^\infty e^{-n(n-1)t} {{\overline{w}}}^n z^n. \end{aligned}$$

Take \(f(z)=\sum _{n=0}^\infty a_n z^n\); then \(F(t,w)=T(t)f(w)\) is given by

$$\begin{aligned} F(t,w)=\sum _{n=0}^\infty a_n e^{-n(n-1)t} w^n. \end{aligned}$$

3 Link between the generator and the numerical range

Let \(n_0\) be a positive integer and let G be an analytic function in the disc (we impose a more general condition on G later).

Assume that A is defined on a dense set of \(H^2\) by

$$\begin{aligned} Af=Gf^{(n_0)}, \end{aligned}$$

where \(f^{(n_0)}\) denotes the derivative of f of order \(n_0\).

If \(G(z)=\sum _{n=0}^{\infty }a_nz^n\), denote by \({\widetilde{G}}\) the analytic function associated with G defined by

$$\begin{aligned} {\widetilde{G}}(z)=a_{n_0} +\sum _{n=1}^{\infty } \left( a_{n_0+n}+\overline{a_{n_0-n}}\right) z^n, \end{aligned}$$

with the convention \(a_j=0\) if \(j<0\).

Note that, for all \(z\in {{\mathbb {T}}}\),

$$\begin{aligned} \mathrm{Re}\left( \overline{z^{n_0}}G(z)\right) =\mathrm{Re}\left( {\widetilde{G}}(z)\right) . \end{aligned}$$
(8)

Consider now the weighted Hardy space \(H^2(\beta )\) where \(\beta \) is the sequence defined by

$$\begin{aligned}\beta _n= {\left\{ \begin{array}{ll} 1 &{} \quad \text{ for } 1 \le n < n_0 \\ \sqrt{\frac{(n-1+n_0)!}{(n-1)!}} &{} \quad \text{ for } n\ge n_0. \end{array}\right. }\end{aligned}$$

In other words, for this particular weight \(\beta \), we have \(H^2(\beta )\subset H^2\) and

$$\begin{aligned} f\in H^2(\beta )\Longleftrightarrow \big \langle f,f\big \rangle _{ H^2(\beta )}:=\sum _{n=0}^{n_0-1}|a_n|^2 +\sum _{n=n_0}^{\infty }\frac{(n-1+n_0)!}{(n-1)!}|a_n|^2<\infty . \end{aligned}$$

Note also that there is a link between the scalar product \(\big \langle , \big \rangle _{ H^2(\beta )}\) and the usual scalar product on \(H^2\) denoted by \(\big \langle , \big \rangle _{H^2}\), namely, for all \(f,g\in H^2(\beta )\), we have

$$\begin{aligned} \big \langle f,g\big \rangle _{ H^2(\beta )} =\big \langle f, z^{n_0}g^{(n_0)}\big \rangle _{H^2}+ \sum _{n=0}^{n_0-1}\frac{f^{(n)}(0)}{n!} \frac{\overline{g^{(n)}(0)}}{n!}. \end{aligned}$$

We can now present a link between a condition on G and the upper boundedness of the numerical range of A acting on \(H^2(\beta )\).

Proposition 3.1

If \(\mathrm{ess}\sup _{w\in {{\mathbb {T}}}}\mathrm{Re}(\overline{w^{n_0}}G(w))\le 0\), and D(A) is dense in \(H^2(\beta )\), where A is defined by \(Af=Gf^{(n_0)}\) then

$$\begin{aligned} \sup \mathrm{Re}\left\{ \big \langle Af,f\big \rangle _{H^2(\beta )} :f\in D(A),\Vert f\Vert _{H^2(\beta )}=1\right\} <\infty . \end{aligned}$$

Proof

Writing \(T_F\) for the Toeplitz operator \(f \mapsto P_{H^2} F f\), we have

$$\begin{aligned} \mathrm{Re}\big \langle Af,f\big \rangle _{H^2(\beta )}= & {} \mathrm{Re}\big \langle Gf^{(n_0)},f\big \rangle _{H^2(\beta )}\\= & {} \mathrm{Re}\big \langle Gf^{(n_0)}, z^{n_0}g^{(n_0)}\big \rangle _{H^2}+ \mathrm{Re}\left( \sum _{n=0}^{n_0-1}\frac{(Gf^{(n_0)})^{(n)}(0)}{n!} \frac{\overline{f^{(n)}(0)}}{n!}\right) \\= & {} \mathrm{Re}\big \langle T_{\overline{z^{n_0}}G}f^{(n_0)},f^{(n_0)}\big \rangle _{H^2}+ \mathrm{Re}\left( \sum _{n=0}^{n_0-1}\frac{(Gf^{(n_0)})^{(n)}(0)}{n!} \frac{\overline{f^{(n)}(0)}}{n!}\right) \\= & {} \big \langle T_{\mathrm{Re}(\overline{z^{n_0}}G)}f^{(n_0)},f^{(n_0)} \big \rangle _{H^2}+ \mathrm{Re}\left( \sum _{n=0}^{n_0-1}\frac{(Gf^{(n_0)})^{(n)}(0)}{n!} \frac{\overline{f^{(n)}(0)}}{n!}\right) \\= & {} \big \langle T_{\mathrm{Re}({\widetilde{G}})}f^{(n_0)},f^{(n_0)} \big \rangle _{H^2}+ \mathrm{Re}\left( \sum _{n=0}^{n_0-1}\frac{(Gf^{(n_0)})^{(n)}(0)}{n!} \frac{\overline{f^{(n)}(0)}}{n!}\right) \\= & {} \mathrm{Re}\big \langle {\widetilde{G}} f^{(n_0)}, f^{(n_0)} \big \rangle _{H^2}+ \mathrm{Re}\left( \sum _{n=0}^{n_0-1}\frac{(Gf^{(n_0)})^{(n)}(0)}{n!} \frac{\overline{f^{(n)}(0)}}{n!}\right) . \end{aligned}$$

It follows that \(\mathrm{ess}\sup _{z\in {{\mathbb {T}}}}\mathrm{Re}({\widetilde{G}}(z))\le 0\) implies that

$$\begin{aligned} \mathrm{Re}\big \langle Af,f\big \rangle _{H^2(\beta )} \le \mathrm{Re}\left( \sum _{n=0}^{n_0-1}\frac{(Gf^{(n_0)})^{(n)}(0)}{n!} \frac{\overline{f^{(n)}(0)}}{n!}\right) \le (n_0!)^2\Vert Gf^{(n_0)}\Vert _{H^2}. \end{aligned}$$

In order to find an upper bound independent of the choice of f, note that we may assume without loss of generality that G and f are polynomials of degree at most \(2n_0\). Moreover since the norm of f in \(H^2(\beta )\) is 1, in particular the Taylor coefficients of f are bounded by 1. It is now clear that there exists a numerical constant \(C>0\) depending only on \(n_0\) and the norm of G in \(H^2(\beta )\), such that

$$\begin{aligned} \mathrm{Re}\big \langle Af,f\big \rangle _{H^2(\beta )} \le C, \end{aligned}$$

for all f in the unit ball of \(H^2(\beta )\). \(\square \)

The following result, which serves as a converse, applies in a large family of weighted Hardy spaces with reproducing kernels. Now the sequence \((\beta _n)_n\) need not depend on the operator A.

Proposition 3.2

Let \((\beta _n)_n\) be a decreasing sequence of positive reals such that \(\liminf _{n\rightarrow \infty }|\beta _n|^{1/n}\ge 1\) and let \(G\in H^2(\beta )\) such that

$$\begin{aligned} \mathrm{ess}\sup _{w\in {{\mathbb {T}}}}\mathrm{Re}(\overline{w^{n_0}}G(w))>0. \end{aligned}$$

Then

$$\begin{aligned} \sup \mathrm{Re}\left\{ \big \langle Af,f\big \rangle _{H^2(\beta )} :f\in D(A),\Vert f\Vert _{H^2(\beta )}=1\right\} =+\infty , \end{aligned}$$

where A is defined on \(D(A)=\{f\in H^2(\beta ):Gf^{(n_0)}\in H^2(\beta )\}\) by \(Af=Gf^{(n_0)}\).

Before proceeding to the proof, we state the following technical lemma which explains the hypothesis on monotonicity of \((\beta _n)_n\).

Lemma 3.3

Let \((\beta _n)_n\) be a decreasing sequence of positive reals. Then for each positive integer N, there exists \(\eta =\eta (N)>0\) such that for all \(z\in {{\mathbb {D}}}\) with \(|w|>1-\delta \), we have

$$\begin{aligned} \sum _{n=0}^N\frac{|w|^{2n}}{\beta _n^2}< \sum _{n=N+1}^\infty \frac{|w|^{2n}}{\beta _n^2}. \end{aligned}$$

Proof

Since \((1/\beta _n)_n\) is increasing, we have

$$\begin{aligned} \sum _{n=0}^N\frac{|w|^{2n}}{\beta _n^2}\le \frac{1}{\beta _N^2}\left( 1+|w|^2+\cdots +|w|^{2N}\right) =\frac{1}{\beta _N^2}\left( \frac{1-|w|^{2N+2}}{1-|w|^2}\right) . \end{aligned}$$

On the other hand, we have

$$\begin{aligned} \sum _{n=N+1}^\infty \frac{|w|^{2n}}{\beta _n^2}\ge \frac{1}{\beta _{N+1}^2}\sum _{n=N+1}^\infty |w|^{2n}=\frac{|w|^{2N+2}}{\beta _{N+1}^2(1-|w|^2)}\ge \frac{|w|^{2N+2}}{\beta _N^2(1-|w|^2)}. \end{aligned}$$

Since \(1-|w|^{2N+2}<|w|^{2N+2}\) is equivalent to \(|w|>(1/2)^{1/(2N+2)}\), for all \(w\in {{\mathbb {D}}}\) such that \(|w|>\eta (N)\) with \(\eta (N)=1-(1/2)^{1/(2N+2)}\), we have

$$\begin{aligned} \sum _{n=0}^N\frac{|w|^{2n}}{\beta _n^2}< \sum _{n=N+1}^\infty \frac{|w|^{2n}}{\beta _n^2}. \end{aligned}$$

\(\square \)

Proof of Proposition 3.2

By hypothesis, there exists \(\delta >0\) and a sequence \((w_k)_k\subset {{\mathbb {D}}}\) such that \(|w_k|\rightarrow 1\) and \(\mathrm{Re}(\overline{w_k^{n_0}}G(w_k))\ge \delta \). Moreover the condition \(\liminf _{n\rightarrow \infty }|\beta _n|^{1/n}\ge 1\) guarantees that the space \(H^2(\beta )\) has reproducing kernels \(k_w\) for all \(w\in {{\mathbb {D}}}\). Now consider the sequence \((\widehat{k_{w_k}})_k\) of normalized reproducing kernels associated with \((w_k)_k\), i.e. \(\widehat{k_{w_k}}=\frac{k_{w_k}}{\Vert k_{w_k}\Vert _{H^2(\beta )}}\). First assume that \(k_{w_k}\in D(A)\). In this case, the remainder of the proof consist in checking that

$$\begin{aligned} \lim _{k\rightarrow \infty }\mathrm{Re}\left( \big \langle A\widehat{k_{w_k}}, \widehat{k_{w_k}}\big \rangle _{H^2(\beta )} \right) =+\infty . \end{aligned}$$

Note that

$$\begin{aligned} \big \langle A\widehat{k_{w_k}}, \widehat{k_{w_k}}\big \rangle _{H^2\left( \beta \right) } =\big \langle G\left( \widehat{k_{w_k}}\right) ^{\left( n_0\right) }, \widehat{k_{w_k}}\big \rangle _{H^2\left( \beta \right) } = \frac{1}{\Vert k_{w_k}\Vert ^2_{H^2\left( \beta \right) }}G\left( w_k\right) k^{\left( n_0\right) }_{w_k}\left( w_k\right) , \end{aligned}$$

where \(k^{(n_0)}_{w_k}(z)=\sum _{n\ge n_0}\frac{n(n-1)\cdots (n-n_0+1)\overline{w_k}^n}{\beta _n^2}z^{n-1}\).

It follows that

$$\begin{aligned} \big \langle Ak_{w_k},k_{w_k}\big \rangle _{H^2(\beta )}=\sum _{n\ge 1}\frac{n(n-1)\cdots (n-n_0+1))G(w_k)\overline{w_k^{n_0}}|w_k|^{2(n-n_0)}}{\beta _n^2}, \end{aligned}$$

and thus

$$\begin{aligned} \big \langle A\widehat{k_{w_k}}, \widehat{k_{w_k}}\big \rangle _{H^2(\beta )} =\frac{\overline{w_k^{n_0}}G(w_k)}{|w_k|^{2n_0}} \frac{\sum _{n\ge n_0}\frac{n(n-1)\cdots (n-n_0+1) |w_k|^{2n}}{\beta _n^2}}{\sum _{n\ge 0}\frac{|w_k|^{2n}}{\beta _n^2}}. \end{aligned}$$

Now, for each positive integer N, take \(\eta (N)\) as in Lemma 3.3, and k sufficiently large so that \(|w_k|>1-\eta (N)\). Then we have

$$\begin{aligned} \frac{\sum _{n\ge n_0}\frac{n(n-1)\cdots (n-n_0+1))|w_k|^{2n}}{\beta _n^2}}{\sum _{n\ge 0}\frac{|w_k|^{2n}}{\beta _n^2}}= & {} \frac{\sum _{n=n_0}^N\frac{n(n-1)\cdots (n-n_0+1)|w_k|^{2n}}{\beta _n^2}}{\sum _{n\ge 0}^N \frac{|w_k|^{2n}}{\beta _n^2} + \sum _{n=N+1}^\infty \frac{|w_k|^{2n}}{\beta _n^2}} \\&+ \frac{ \sum _{n=N+1}^\infty \frac{n(n-1)\cdots (n-n_0+1)|w_k|^{2n}}{\beta _n^2}}{\sum _{n\ge 0}^N \frac{|w_k|^{2n}}{\beta _n^2} + \sum _{n=N+1}^\infty \frac{|w_k|^{2n}}{\beta _n^2}}\\\ge & {} \frac{(N+1)N\cdots (N+1-n_0+1)\sum _{n=N+1}^\infty \frac{|w_k|^{2n}}{\beta _n^2}}{2\sum _{N+1}^\infty \frac{|w_k|^{2n}}{\beta _n^2}}\\= & {} \frac{(N+1)N\cdots (N+1-n_0+1)}{2}. \end{aligned}$$

Therefore, for k sufficiently large (so that \(|w_k|>1-\eta (N)\)), we get

$$\begin{aligned} \mathrm{Re}\left( \big \langle A\widehat{k_{w_k}}, \widehat{k_{w_k}}\big \rangle _{H^2(\beta )} \right) \ge \frac{(N+1)N\cdots (N+1-n_0+1)}{2|w_k|^{2n_0}}\mathrm{Re}(\overline{w_k^{n_0}}G(w_k)). \end{aligned}$$

Since \(\mathrm{Re}(\overline{w_k^{n_0}}G(w_k))\ge \delta \) and since \(|w_k|\) tends to 1, we get the desired conclusion.

If \(k_{w_k}\) is not in D(A), the conclusion follows from similar calculation, considering the sequence of polynomials \((k^M_{w_k})_{M\ge 0}\) defined by

$$\begin{aligned} k^M_{w_k}=\sum _{n=0}^M\frac{\overline{w_k^n}}{\beta _n^2}z^n, \end{aligned}$$

which belongs to D(A) and tends to \(k_{w_k}\) in \(H^2(\beta )\). \(\square \)

Corollary 3.4

For A to generate a \(C_0\) semigroup of quasicontractions on \(H^2(\beta )\) it is necessary and sufficient that \(\mathrm{ess}\sup _{w\in {{\mathbb {T}}}}\mathrm{Re}(\overline{w^{n_0}}G(w))\le 0\) and there is a \(\lambda >0\) such that \(A-\lambda I\) is invertible in \(H^2(\beta )\). Moreover, such a semigroup is holomorphic if and only if there is a \(\beta \in (0,\pi /2)\) such that \(\mathrm{ess}\sup _{w\in {{\mathbb {T}}}}\mathrm{Re}(e^{\pm i\beta }\overline{w^{n_0}}G(w))\le 0\).

Proof

The first part follows from Propositions 3.1 and 3.2 using the Lumer–Phillips theorem [14, p. 14]. The characterization of holomorphy follows from the complex version of the Lumer–Phillips theorem as in [5, Prop. 2.2]. \(\square \)