A class of quasicontractive semigroups acting on Hardy and weighted Hardy spaces

This paper studies semigroups of operators on Hardy and Dirichlet spaces whose generators are differential operators of order greater than one. The theory of forms is used to provide conditions for the generation of semigroups by second order differential operators. Finally, a class of more general weighted Hardy spaces is con-sidered and necessary and sufﬁcient conditions are given for an operator of the form f (cid:2)→ G f ( n 0 ) (for holomorphic G and arbitrary n 0 ) to generate a semigroup of quasicontractions.


Introduction
The theory of semigroups of weighted composition operators on Hardy and Dirichlet spaces of the unit disc D is well understood (see [1,[3][4][5][6]12,15,16]), and these have generators that are first order differential operators of the form A : f → 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 → G f (n 0 ) to generate a semigroup of quasicontractions.
Take (β n ) n≥0 a sequence of positive real numbers. Then H 2 (β) is the space of analytic functions c n z n in the unit disc that have finite norm The case β n = 1 gives the usual Hardy space H 2 ; also, the case β 0 = 1 and β n = √ n for n ≥ 1 provides the Dirichlet space D, which is included in H 2 (D).
With an extra condition on (β n ) n , the Hilbert space H 2 (β) is also a reproducing kernel space, i.e., for all w ∈ D, there exists a function k w ∈ H 2 (β) such that for all f ∈ H 2 (β) (see p. 19 in [7] and p. 146 in [13]). More precisely, if (β n ) n is such that it follows that H 2 (β) is a reproducing kernel Hilbert space and In fact (1) is also equivalent to the more explicit condition lim inf(β n ) 1/n ≥ 1.
We recall that H ∞ 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.
Note that if f (z) = z n , then (A f )(z) = −n(n − 1)z n . It follows easily that A is the generator of the semigroup (T (t)) t≥0 , where ∞ n=0 a n z n = ∞ n=0 a n e −n(n−1)t z n .

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 × V → C a form, that is a continuous, sesquilinear and coercive mapping, i.e. there exists α > 0 such that, for all u ∈ V , Then we take H another complex Hilbert space such that there is an embedding j : V → H continuous and dense range.
Then the linear operator A is well-defined, linear, densely-defined, with A : The link between a and A is the following.
for all t in θ .
We now present an application of the above theory from forms to semigroups.
where ε 1 is a positive numerical constant. Then define a by Using the Cauchy-Schwarz inequality, a is continuous and obviously sesquilinear. It remains to check that a is j-elliptic.
Note that Moreover, once more using the Cauchy-Schwarz inequality, we have Re Now for G 3 ∞ < 2 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 A f = g when Indeed, one can easily check this identity for k and h equal to powers of z.
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]).
Then −A generates a holomorphic C 0 -semigroup of quasicontractions.
An easy example here is given by The proof of Theorem 2.2 leads to the following particular cases.
If moreover there exists ε 2 > 0 such that 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 A f (z) = z(zG 1 (z) f (z)) for which we can conclude that −A generates a holomorphic C 0semigroup of quasicontractions (T (z)) z∈ θ with T (z) ≤ e wRe(z) for all w > 0. This implies that T is a holomorphic C 0 -semigroup of contractions.

Recall that an operator B is bounded relative to another operator A if D(A) ⊆ D(B)
and there are constants a, b > 0 such that We write a 0 = inf{a > 0 : ∃b such that (6) holds}, and call this the A-bound of B.
To see an application of this result, let us take A defined by A f = −z 2 f for f ∈ D(A) ⊂ H 2 , which is diagonalisable with Ae n = −n(n−1)e n , where e n (z) = z n for n = 0, 1, 2, . . . Now (λI − A) −1 e n = e n λ + n(n − 1) .

Heat kernels
It is well known (see, e.g., [10]) that many parabolic partial differential equations on R n can be solved in terms of a (generalized) heat kernel. The simplest example is defined on L 2 (R) by giving a semigroup whose generator is the closure of the Laplacian f → f .
Let H be a reproducing kernel Hilbert space of analytic functions on D. For a semigroup (T (t)) t≥0 on H with generator f → n 0 k=0 G k f (k) , where the G k are analytic in D, define the associated kernel K (t, z, w) on R + × D × D by where k w is the reproducing kernel for H . That is, K satisfies the equation Proposition 2.7 Suppose that T (t) f (w) = F(t, w); then 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).

Theorem 2.8
If the equation n 0 k=0 G n ϕ (k) = λϕ has a normalized basis (ϕ n ) of eigenvectors in H , with eigenvalues (λ n ), such that Re λ n ≤ 0 for all n, then Proof By a simple calculation f (w) = f, K (0, z, w) z for f ∈ H , and so K (0, z, w) = k w (z). Moreover, the hypotheses on (ϕ n ) and λ n easily imply that (7) holds.

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 where f (n 0 ) denotes the derivative of f of order n 0 . If G(z) = ∞ n=0 a n z n , denote by G the analytic function associated with G defined by G(z) = a n 0 + ∞ n=1 a n 0 +n + a n 0 −n z n , with the convention a j = 0 if j < 0.
Note that, for all z ∈ T, Consider now the weighted Hardy space H 2 (β) where β is the sequence defined by In other words, for this particular weight β, we have H 2 (β) ⊂ H 2 and Note also that there is a link between the scalar product , H 2 (β) and the usual scalar product on H 2 denoted by , H 2 , namely, for all f, g ∈ H 2 (β), we have 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 (β).
Proof Writing T F for the Toeplitz operator f → P H 2 F f , we have It follows that ess sup z∈T Re( G(z)) ≤ 0 implies that 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 (β) 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 (β), such that for all f in the unit ball of H 2 (β).
The following result, which serves as a converse, applies in a large family of weighted Hardy spaces with reproducing kernels. Now the sequence (β n ) n need not depend on the operator A. Before proceeding to the proof, we state the following technical lemma which explains the hypothesis on monotonicity of (β n ) n .