Composition Semigroups on Weighted Bergman Spaces Induced by Doubling Weights

We prove that composition semigroups are strongly continuous on weighted Bergman spaces with doubling weights. Point spectra and compact resolvent operators of infinitesimal generators of composition semigroups are characterized.


Introduction
Let H(D) denote the space of analytic functions in the unit disc D � z ∈ C: |z| < 1 { }. For a nonnegative function ω ∈ L 1 ([0, 1)), the extension to D, defined by ω(z) � ω(|z|) for all z ∈ D, is called a radial weight. For 0 < p < ∞ and a radial weight ω, the weighted Bergman space A p ω consists of f ∈ H(D) such that where dA(z) � dx dy/π is the normalized Lebesgue area measure on D. As usual, A p α stands for the classical weighted Bergman space induced by the standard radial weight For a radial weight ω, write ω(z) � 1 |z| ω(s)ds for all z ∈ D. In this paper, we always assume ω(z) > 0, for otherwise A p ω � H(D) for each 0 < p < ∞. A weight ω belongs to the class D if there exists a constant C � C(ω) ≥ 1 such that ω(r) ≤ Cω(1 + r/2) for all 0 ≤ r < 1. For more knowledge about those Bergman spaces, see [1][2][3][4] and the reference therein.
A family (φ t ) t ≥ 0 of analytic self-maps of the unit disk D in the complex plane C is said to be a semigroup if the following conditions hold: For any nontrivial semigroup (φ t ) t ≥ 0 , there exist a point b ∈ D and an analytic function P: D↦C with ReP ≥ 0 such that Refer [5] for the details. Representation (3) is unique, and the point b is said to be the Denjoy-Wolff point of Notice that each semigroup (φ t ) t ≥ 0 gives rise to a semigroup (C t ) t ≥ 0 consisting of composition operators on H(D), the set of analytic functions on D, where Given a semigroup (φ t ) t ≥ 0 and a Banach space X of analytic functions on D, we say that (φ t ) t ≥ 0 generates a strongly continuous composition operator on X if C t is bounded on X and where I is the identity map on X. e infinitesimal generator of a strongly continuous semigroup (C t ) t ≥ 0 on a Banach space X is the operator which is densely defined for every x in the domain Refer [6] for more information about operator semigroup. In 1978, Berkson and Porta [5] initially studied the strong continuity of semigroups of composition operators acting on the classical Hardy space H p (D). ey proved that [7,8]. Moreover, he showed that the infinitesimal generator Γ of a semigroup of composition operators on these spaces is of the form Γf � Gf ′ with a certain domain. Refer [9][10][11][12][13][14][15][16] for more results of composition semigroups on other various spaces.
In this paper, we consider composition semigroups on weighted Bergman spaces A p ω with ω ∈ D. We will show that every e corresponding infinitesimal generator Γ of (C t ) t ≥ 0 and its point spectrum can also be identified. In addition, if the Denjoy-Wolff point of (φ t ) t ≥ 0 belongs to D, then we can also characterize the compactness of resolvent operator R(λ, Γ) of Γ, provided λ belongs to the resolvent set of Γ.
roughout the paper, the symbol A ≈ B means that A≲B≲A. We say that A≲B if there exists a constant C such that A ≤ CB.

Strongly Continuous Composition Semigroup
We need more information about (φ t ) t ≥ 0 before presenting our results. Assume a semigroup (φ t ) t ≥ 0 consisting of analytic selfmaps of D with infinitesimal generator G and Denjoy-Wolff point b. In general, (φ t ) t ≥ 0 can be classified into two classes: b ∈ D and b ∈zD, the boundary of D. In particular, there exists a unique univalent function h: D ⟶ C, called Koenigs function, such that Lemma 1 shows that every composition operator is bounded on A p ω if 0 < p < ∞ and ω ∈ D. e proof can be easily obtained by a simple combination of eorem 15 and (4.7) in [17].
en, the composition C φ is bounded on A p ω . Moreover, there exist constants η � η(ω) > 1 and C � C(η, ω, p) such that Now, we are ready to show our results. For f ∈ H(D) and 0 < r < 1, set Theorem 1. Let 1 ≤ p < ∞ and ω ∈ D. Suppose (φ t ) t ≥ 0 is a semigroup of analytic self-maps of D with infinitesimal generator G. en, the induced composition semigroup (C t ) t ≥ 0 defined in (4) is strongly continuous on A p ω with infinitesimal generator Γ: on its domain Moreover, (C t ) t ≥ 0 is uniformly continuous on A p ω if and only if (φ t ) t ≥ 0 is trivial.

Proof.
Since ω ∈ D, the polynomials are dense in A p ω . erefore, for any f ∈ A p ω , there exists a sequence of polynomials P n such that lim n⟶∞ ‖f − P n ‖ A p ω � 0. It follows triangle inequality that According to Lemma 1, we know that sup t∈[0,1] ‖C t ‖ < ∞. erefore, to prove lim t⟶0 + ‖C t f − f‖ A p ω � 0, it suffices to prove lim t⟶0 + ‖C t P − P‖ A p ω � 0 for each polynomial P. Equivalently, we only need to show that for each n ≥ 0, lim t⟶0 + ‖(φ t ) n − e n ‖ A p ω � 0, where e n (z) � z n , while it can be easily obtained by Lebesgue-dominated convergence theorem. us, (C t ) t ≥ 0 is strongly continuous on A p ω . By definition, the domain of Γ is 2 Journal of Mathematics Since for a fixed 0 < r < 1, the well-known inequality Convergence in the norm of A p ω implies the pointwise convergence. erefore, for every z ∈ D, (20) On the other hand, for λ ∈ ρ(Γ), the resolvent set of Γ, we have where R(λ, Γ) � (λI − Γ) − 1 is the resolvent operator of Γ.
To the end, we may consider polynomials e n (z) � z n . en, Γ(e n ) � nGe n− 1 , and taking n � 1, we see that G ∈ A p ω . Since Γ is bounded on A p ω , for n ≥ 1, we have ‖Γe n ‖≲‖e n ‖, that is,  Proof. (i) By (7), Now, suppose f ∈ H(D) and λ ≠ 0 such that Pick r such that |b| < r < 1 and f has no zeros on |z| � r. We have dz.

(25)
From this and the argument principle, it follows that λ/G ′ (b) � k, a nonnegative integer, which shows the first part of (i). Also, notice that the differential equation is shows the second part of (i).
e proof is complete.

Resolvent Operator
In [18], Siskakis characterized the compactness of R(λ, Γ) on the Hardy space H p and the weighted Bergman space A p α if the Denjoy-Wolff point of (φ t ) t ≥ 0 is in the interior of D. In this section, we will consider the compactness of R(λ, Γ) on A p ω with ω ∈ D.
To prove the main result in this section, we need the following lemma, which characterizes the boundedness and