Spectrums and uniform mean ergodicity of weighted composition operators on Fock spaces

For holomorphic pairs of symbols $(u, \psi)$, we study various structures of the weighted composition operator $ W_{(u,\psi)} f= u \cdot f(\psi)$ defined on the Fock spaces $\mathcal{F}_p$. We have identified operators $W_{(u,\psi)}$ that have power bounded and uniformly mean ergodic properties on the spaces. These properties are described in terms of easy to apply conditions relying on the values $|u(0)|$ and $|u(\frac{b}{1-a})|$ where $ a$ and $b$ are coefficients from linear expansion of the symbol $\psi$. The spectrum of the operators are also determined and applied further to prove results about uniform mean ergodicity.


Introduction
We denote by H(C) the space of analytic functions on the complex plane C. For pairs of functions (u, ψ) in H(C), the weighted composition operator W (u,ψ) is defined by W (u,ψ) f = u · f (ψ), f ∈ H(C). The operator generalizes both the composition C ψ and multiplication M u operators since it can be factored as W (u,ψ) = M u C ψ . Weighted composition operators have been a subject of intense studies in the last several years partly because they found applications in the description of isometries on spaces of analytic functions; see the monographs [10,11] for detailed accounts. For studies on the various properties of the operators, for example on the classical Fock spaces F p , one may consult the works in [15,24,25,27] and the references therein. Recall that F p are spaces consisting of all entire functions f for which where dA is the Lebesgue area measure on C. For each function f ∈ H(C), the subharmonicity of |f | p implies that the local point estimate By definition of the norm, the estimate in (1.2) is valid for p = ∞ as well. The space F 2 is a reproducing kernel Hilbert space with kernel function K w (z) := e wz and normalized kernel k w := K w −1 2 K w . A straightforward calculation shows that k w belongs to all the Fock spaces F p with a unit norm k w p = 1 for all w ∈ C.
The rest of the manuscript is organized as follows. In Section 2, we study the power bounded property of the operators. As stated in Theorem 2.1 and Theorem 2.3, these properties are described in terms of simple to apply conditions which are merely based on the values of the numbers |u(0)| or |u( b 1−a )| where the constants a and b are from the linear expansion of the symbol ψ(z) = az + b. The proofs of the results are presented in Subsections 2.1 and 2.2. In Section 3, we identify the spectra of the operators on F p for all 1 ≤ p ≤ ∞; see Theorem 3.1 whose proof is given in Subsection 3.1. Section 4 contains several results on the uniform mean ergodic properties of the operators.
We conclude this section with a word on notation. The notion U(z) V (z) (or equivalently V (z) U(z)) means that there is a constant C such that U(z) ≤ CV (z) holds for all z in the set of a question. We write U(z) ≃ V (z) if both U(z) V (z) and V (z) U(z).

Power bounded W (u,ψ)
We start this section by recalling certain definitions related to dynamics of an operator. Let T be a bounded operator on a Banach space X . Then we set the operator T n as the n-th iterate of T ; T n = T • T • ... • T n-times and T 0 = I where I is the identity map on X . The operator T is said to be power bounded on X if sup n∈N T n < ∞. Obviously, any operator with norm at most 1 is power bounded. The notion of power boundedness or estimating T n plays an important roll in the study of numerical stability of initial value problems. If T m , n ∈ N denotes its the n th Cesàro means, then T is called mean ergodic if there exists a bounded operator P on X such that for each f in X and uniformly mean ergodic if the pointwise convergence above is uniform; A straightforward simplification gives that for each n ∈ N, the relation 1 n T n = T n − n − 1 n T n−1 holds where we set T 0 = I as the identity operator on X . This immediately implies that if T is mean ergodic, then 1 n T n x → 0 as n → ∞ for all x ∈ X . Similarly, 1 n T n → 0 whenever T is uniformly mean ergodic. A number of authors have studied ergodicity of operators on various functional spaces; see for example [1,3,5,6]. The monographs [14,28] provides basic information on ergodic theory. Inspired by all these works, the authors and J. Bonet in [26] studied the mean ergodicity of composition operators acting on generalized Fock spaces, and concluded that all bounded composition operators on Fock spaces F p are power bounded whenever 1 ≤ p ≤ ∞. In this section, we show that this conclusion is no longer true in general for the weighted composition operators W (u,ψ) . It is found that W (u,ψ) is power bounded only when the weight function u satisfies an interesting point value condition as precisely stated in the next two main theorems and proposition.
The study of the dynamics of an operator is related to the study of its iterates. For f ∈ H(C), a simple argument shows that the image of f under the iterates of W (u,ψ) has the form for each n ∈ N and ψ 0 = I the identity map on C. The equations in (2.1) will be repeatedly used in the sequel. We now state the main results on power boundedness. Depending on whether |a| = 1 or |a| < 1, we give two main results in Theorem 2.1 and Theorem 2.3.

(2.2)
It is interesting that we have an easy to apply equivalent conditions for the power boundedness of the weighted composition operators. Part (iii) of the condition is also independent of underlying space or the exponents p. In addition, by setting n = 1, the theorem provides a simple expression for the norm of the operators namely that We recall that a bounded linear operator is a contraction when its norm is bounded by 1. In view of this, we may add one more equivalent condition to the above list in the theorem, namely that W (u,ψ) is power bounded on F p if and only if it is a contraction. It should be also noted that for the case a = 1 and |a| = 1, the above conditions are also equivalent to u b 1−a ≤ 1 since an application of Lemma 2.4 below implies where ℜ denotes the real part of the given complex number. This inspires us to ask whether a similar condition works for the remaining case namely that when |a| < 1. In this case, as will be explained later, the powers of weighted composition operators are again weighted composition operators. This together with the relations in (2.6) and (2.7) ensure that the following necessary and sufficient conditions hold whenever |a| < 1.
When p → ∞, the right-hand side in (2.4) tends to 1. Thus, the condition in (2.3) is both necessary and sufficient for W (u,ψ) to be power bounded on the space F ∞ . In particular when W (u,ψ) is compact, we record our next main result which holds true on all the spaces F p . Theorem 2.3. Let 1 ≤ p ≤ ∞, u, ψ ∈ H(C) and W (u,ψ) be bounded on F p , and ψ(z) = az + b, with |a| < 1. Let u be non-vanishing and W (u,ψ) be compact. Then the following statements are equivalent.
(i) W (u,ψ) is power bounded on F p ; (ii) ( u n p ) n is a bounded sequence; (iii) |u b 1−a | ≤ 1. As in Theorem 2.1, part (iii) of the condition is simple to apply and independent of the exponents p. Observe that from the two theorems above, it is easy to see that a bounded composition operator C ψ is always power bounded while the multiplication operator M u is not in general; see Corollary 4.7.
To prove the results, we need to make some preparations. The bounded and compact weighted composition operators on Fock spaces were characterized first in terms of Berezin-type integral transforms in [24,25,27]. Later, Le [15] considered the Hilbert space F 2 setting and obtained a simpler condition namely that W (u,ψ) is bounded on  (2.5) He further proved that (2.5) implies ψ(z) = az + b with |a| ≤ 1. In [22], T. Mengestie and M. Worku proved that the Berezin-type integral condition used to describe the boundedness of generalized Volterra-type integral operators V (g,ψ) on the Fock spaces F p is equivalent to a simple condition as in (2.5). Because of the Littlewood-Paley type description of the Fock spaces, by simply replacing |g ′ (z)|/(1 + |z|) by |u(z)| in the results there, it has been known that (2.5) in fact describes the bounded weighted composition operators on all the spaces F p , 1 ≤ p < ∞, with norm bounds For p = ∞, the corresponding relation holds with equality, As indicated in [15], an interesting consequence of (2.5) is that if |a| = 1, then a simple argument with Liouville's theorem gives that the weight function u has the form u(z) = u(0)K −ab (z). This representation of u will play an important roll in the rest of the paper. Thus, we may formulate it as a lemma for the purpose of easy further referencing.
Lemma 2.4. Let 1 ≤ p ≤ ∞, u, ψ ∈ H(C) and W (u,ψ) be bounded on F p , and hence ψ(z) = az + b, |a| ≤ 1. If |a| = 1, then It should be noted that by condition (2.5) it is possible for W (u,ψ) = M u C ψ to be bounded even if both the factors C ψ and M u are unbounded. The functions u(z) = z and ψ(z) = z + 1 provides such an example.
Similarly, compactness of W (u,ψ) has been described by the fact that ψ(z) = az + b, |a| ≤ 1 and |u(z)|e The later condition implies that |a| < 1 but not conversely. Very recently, Carroll and Gilmore [7], used the idea of order of analytic function and proved the following analogues result.
Lemma 2.5. Let 1 ≤ p ≤ ∞, u, ψ ∈ H(C) and ψ(z) = az + b, |a| < 1, and assume that u is non-vanishing. Then W (u,ψ) is compact on F p if and only if u has the form u(z) = e a 0 +a 1 z+a 2 z 2 (2.8) for some constants a 0 , a 1 , a 2 such that |a 2 | < 1−|a| 2 2 . Next, we consider the following key necessary conditions for power bounded W (u,ψ) . The lemma gives a good restriction on the growth of the sequence ( u n p ) n and the value |u(z 0 )| where z 0 is a fixed point of ψ.
Proof. (i). Since the constant function 1 belongs to the spaces F p with 1 p = 1, using the pointwise estimate in (1.2) To prove (ii), for p = ∞ arguing as above we have Taking the supremum with respect to first with z and then with n give the required assertion. On the other hand, if p < ∞, then from which the conclusion follows again.
The next simple lemma will be crucial in the proof of Theorem 2.3.
We are now ready to give the proofs of the previous two main results.
for each z ∈ C. If a = 1, then ψ j (z) = z + jb and using Lemma 2.4, It follows that for all z ∈ C. Considering (2.11) and applying the estimate in (2.10) at z = −nb and the conclusion follows after taking this in (2.10) again. It remains to prove (iii) implies (i). First observe that for each n ∈ N, the operator W n (u,ψ) itself is a weighted composition operator and W n Thus, for |a| = 1, we apply (2.6) and obtain the norm Our next task is to simplify (2.12). If a = 1, then the representation in (2.11) implies from which the statement follows. Next, assume a = 1 and |a| = 1.

2.2.
Proof of Theorem 2.3. The statement (i) implies (ii) follows from Lemma 2.6 again. Assuming (ii), we proceed to show that (iii) holds. Using (1.2) we estimate which together with Lemma 2.4 and (2.15) gives statement (iii). Next, we prove (iii) implies (i). If p = ∞, applying the relation in (2.7) for the weighted composition operator W (un,ψ n ) we get (2.17) Now taking this into account and the fact that W n (u,ψ) = W (un,ψ n ) , the corresponding notation in (2.5) becomes M(u n , ψ n ) = sup z∈C |u n (z)|e where c n is the real part of the expression Now to estimate the supremum in (2.18), we claim that Observe that the inequality holds if and only if This follows immediately from Lemma 2.7 as |a 2 | < 1−|a| 2 2 .
It follows from this and (2.18) that M(u n , ψ n ) e cn .
On the other hand, since |a| < 1 From which we arrive at the claim. Here note that since W (u,ψ) is bounded, the multiplier u belongs to F p for all p.
If a = 0, then applying the local point estimate in (1.1), Observe that if w ∈ D a n z + b(1−a n ) 1−a , 1 , then which holds true if and only if 1 |a| n ≥ z − w a n − b(1 − a n ) a n (1 − a) .
Thus, w belongs to the disk D a n z + b(1 − a n ) 1 − a , 1 if and only if z belongs to D w a n − b(1 − a n ) 1 − a , 1 |a| n .
Making use of this and Fubini's theorem in (2.20) Using Lemma 2.5 and simplifying like the case for p = ∞, we get for all z ∈ C. Since W (u,ψ) is compact and |a 2 | < 1−|a| 2 2 , it follows that and C e p 2 a n z+ b(1−a n ) where we used the fact that sup n∈N t n < ∞ and inf n {q n − |p n |} > 0. Hence, W n (u,ψ) f p e cn f p and the conclusion follows as in the last part of p = ∞, and completes the proof of the theorem.

The spectrum of W (u,ψ)
In this section, we study the spectral of weighted compositions operators on all the Fock spaces F p , 1 ≤ p ≤ ∞. Let T be a bounded linear operator on a Banach space X . Then the spectrum σ(T ) of T is the set {λ ∈ C : T − λI is not invertible} where I is the identity operator on X . The spectrum σ(T ) is always a non-empty compact and closed subset of the disk centered at the origin and of radius T . It has been well-known that the spectrum of an operator plays a vital roll in the study of its dynamical properties; see [12]. Our next result will be used to prove mean ergodic results in the next section apart from being interest of its own. Theorem 3.1. Let 1 ≤ p ≤ ∞, u, ψ ∈ H(C) and W (u,ψ) be bounded on F p and hence ψ(z) = az + b with |a| ≤ 1. Then if (i) W (u,ψ) is compact and hence |a| < 1, then  1 − a))a m . In this case, the spectrum contains finite number of points only when a is a root of unity. Next, for a = 1, the necessity of the condition that |u(b/(1 − a))| ≤ 1 in Theorem 2.1 and Theorem 2.3 can be easily deduced using Theorem 3.1 since the spectrum of a power bounded operator is always a subset of the closed unit disc. Here for |a| = 1 and a = 1, note that u(0)e a|b| 2 a−1 = u b 1−a . Another application of Theorem 3.1 will be given in Section 4. It is known that the spectrum plays an essential roll in the study of theory of semigroups of linear operators [9] as well.
3.1. Proof of Theorem 3.1. (i). Let W (u,ψ) is compact and hence |a| < 1. Here our proof is based on an argument that goes back to [13]. We set z 0 = b/(1 − a) and plan to show that the range of W (u,ψ) − a n u(z 0 )I fails to contain the complex polynomial z n . Setting n = 1 and arguing in the direction of contradiction, assume that there exists an f ∈ F p such that If u(z 0 ) = 0 or a= 0, then au(z 0 ) = 0 and belongs to the spectrum. Thus, we may assume that z 0 is not in the zero set of u and a = 0. First assume that z 0 = 0. Then taking z = 0 in (3.2), we obtain that f (0) = 0. On the other hand, differentiating both side of equation (3.2) and setting again z = 0, we obtain which results the contradiction 0 = 1. Similarly for n > 1, differentiating both side of the equation u(z)f (ψ(z)) − a n u(z 0 )f (z) = z n repeatedly and eventually setting z = 0, we obtain f (m) (0) = 0 for all m < n while for m = n we get again the contradiction 0 = n!.
If z 0 = 0, then we may set ψ 1 (z) = az, and observe that ψ 1 (0) = 0 and u 1 (0) = u(z 0 ). A straightforward calculation shows that It follows that the weighted composition operators W (u 1 ,ψ 1 ) and W (u,ψ) are similar and have the same spectrum, and our conclusion follows from the first case discussed above. Therefore, the set in the right-hand side of (3.1) in this case is contained in the spectrum. Conversely, if |a| < 1, then W (u,ψ) is compact and its spectrum contains only zero and eigenvalues. Thus, we consider a nonzero eigenvalue λ ∈ σ(W (u,ψ) and show that it is of the form u(z 0 )a n for some positive integer n. If f is a corresponding nonzero eigenvector, then for all z in C. If f has no zero at z 0 , then (3.3) implies u(z 0 ) = λ and hence λ = a 0 u(z 0 ). On the other hand, if f has zero at z 0 of order m, we may write where g(z 0 ) = 0. Then substituting f by this in The argument in the proof of part (ii) is divided into three cases depending on the values of a and b. Case 1. Let |a| = 1 and a = 1. For simplicity we first set ψ z 0 (z) = z − z 0 and claim that the weighted composition operator induced by (k z 0 , ψ z 0 ) is an isometric bijective map on F p with inverse W (k −z 0 ,ψ −1 z 0 ) . To this claim, for every f ∈ F p for all 1 ≤ p < ∞ which also holds true for p = ∞. This shows that the operator is a linear isometry and hence satisfies the injectivity condition W −1 (kz 0 ,ψz 0 ) W (kz 0 ,ψz 0 ) = I. On the other hand, for each f ∈ F p which also shows that W (kz 0 ,ψz 0 ) W −1 (kz 0 ,ψz 0 ) = I, and hence the claim. Next, using z 0 = b/(1 − a) and Lemma 2.4 for every f ∈ F p we compute where C ψ 0 the composition operator induced by the symbol Ψ 0 (z) = az. This shows that W (u,ψ) is similar to the composition operator, up to a multiple, C Ψ 0 .
Thus, σ(W (u,ψ) ) = u(0)e |b| 2 1−a σ(C Ψ 0 ). Using the spectrum of C Ψ 0 from Theorem 2.6 of [23] and observing that (1 − a) −1 = a/(1 − a) when |a| = 1 and a = 1, we arrive at the desired conclusion. Case 2. Let a = 1 and b = 0. Applying Lemma 2.4, The weighted composition operator W (k −b ,ψ) is unitary. Recall that the spectrum of a unitary operator lies on the unit circle T. We claim that the spectrum of To prove the claim, for any nonzero w ∈ C and f ∈ F p we have which shows that W (k −b ,ψ) is similar to e 2iℑ(wb) W (k −b ,ψ) for any w ∈ C. Since b = 0, and e 2iℑ(wb) is unimodular, and the spectrum of a unitary operator lies on the unit circle T, it follows that the whole unit circle constitutes the spectrum. Therefore, combining this with (3.5) σ(W (u,ψ) ) = u(0)e . and completes the proof. Case 3. Let a = 1 and b = 0. In this case, the operator W (u,ψ) reduces to the multiplication operator M u where its spectrum has been already identified in Lemma 2.3 of [21].

Uniformly mean ergodic W (u,ψ)
Having identified conditions under which W (u,ψ) is power bounded, we next turn our attention to the mean and uniformly mean ergodic properties of W (u,ψ) on F p . We may first state the following result about compact weighted composition operators on the spaces. Proposition 4.1. Let 1 ≤ p ≤ ∞ and W (u,ψ) be a compact power bounded operator on F p , and hence ψ(z) = az +b such that |a| < 1. Let u be non-vanishing on C . Then W (u,ψ) is uniformly mean ergodic.

Proof. A result of Yosida and Kakutani ([29] Theorem 4 and Corollary on page
204-205 and Theorem 2.8 in [14]) implies that every compact power bounded operator on a Banach space is uniformly mean ergodic. Thus, W (u,ψ) is uniformly mean ergodic.
Theorem 4.2. Let 1 ≤ p < ∞ and W (u,ψ) be a compact power bounded operator on F p , and hence ψ(z) = az + b such that |a| < 1. Let u be non-vanishing on C such that u(z 0 ) = 1 where z 0 = b/(1 − a). Then u(ψ j (z)). For each f ∈ F p we claim that Observe that (4.2) implies (4.1). Thus, we proceed to prove the claim by considering two different cases. Let p < ∞. Since W (u,ψ) is power bounded, by Lemma 2.6, u n belongs to F p for all n ∈ N. On the other hand, using the representation of u n in (2.17) and applying Theorem 2.3 and Lemma 2.7 (1 − a n )b 1 − a a n , and letting n → ∞ The compactness condition Moreover, by continuity, and since W (u,ψ) is power bounded, there is a constant α > 0 such that for every

Consequently, we have
Applying Lebesgue dominated convergent theorem on the sequence we get that Proof. For this, we consider the subspace F 0 of F ∞ defined by This subspace is closed in F ∞ and it contains polynomials. Moreover, the polynomials are dense in F 0 , and F ∞ is canonically isomorphic to the bidual of F 0 ; see for details in [4]. We proceed to show first that (4.2) holds for each f ∈ F 0 . It is easy to see that as n → ∞ ψ n (z) = a n z + z 0 (1 − a n ) → z 0 uniformly on compact subsets of C.
Next, we show that u n → u ∞ uniformly on compact subset of C also. Since Applying Lebesgue dominated convergence theorem, for z ∈ K, where K is compact subset of C, From this we have uniformly on the compact subsets of C. That is, for each compact set K in C, as n → ∞.
Note that the last inequality above holds since W (u,ψ) is power bounded the supremum above is uniformly bounded. That is sup n sup z∈C |u n (z)|e 1 2 |a n z+z 0 (1−a n )| 2 −|z| 2 |u(z 0 )| n ≤ 1.
The preceding results assure that W (u,ψ) with ψ(z) = az + b, |a| < 1 is always uniformly mean ergodic whenever it is compact and power bounded. Now, we consider the case when ψ(z) = az + b and |a| = 1. Note that power boundedness in this case implies that either |u(0)| = e − |b| 2 2 or |u(0)| < e − |b| 2 2 . In 1939, Lorch [19] proved that every power bounded operator on a reflexive Banach space is mean ergodic. The same result was latter obtained in reflexive Frechet spaces [1]. Accordingly, as the spaces F p are reflexive for all 1 < p < ∞, every power bounded W (u,ψ) is mean ergodic. Thus, for such spaces we will consider conditions under which the ergodicity becomes uniform. (ii) Let 1 ≤ p ≤ ∞,and ψ(z) = az with |a| = 1. If both u(0) and a are roots of unity, then W (u,ψ) is uniformly mean ergodic on F p .
By a result in [26], the composition operator C ψ is not uniformly mean ergodic on F ∞ whenever |a| = 1. Now the weight function u makes it possible to enrich uniformity by taking the value |u(0)| smaller. (ii) By assumption there exist numbers m, N ∈ N such that a N = 1 = u(0) m . Consider the smallest positive integer N 0 ≤ mN such that a N 0 = u(0) N 0 = 1. In this case the sequence W n u,ψ is periodic with period N 0 . Any n ∈ N can be written in the form of n = N 0 l + j for some l ∈ N and j = 0, 1, 2, ..., N 0 − 1. Thus and completes the proof.
Next, we consider the cases when the uniform ergodicity fails. We may first recall that a Banach space X is a Grothendieck space if every sequence (x n ) in X ′ which is convergent to 0 for the weak topology σ(X ′ , X) is also convergent to 0 for the weak topology σ(X ′ , X ′′ ). The space X has the Dunford-Pettis property if for any sequence (x n ) in X which is convergent to 0 for the weak topology σ(X, X ′ ) and any sequence (x ′ n ) in X ′ which is convergent to 0 for the weak topology σ(X ′ , X ′′ ) one gets lim n→∞ x On the other hand, for each f ∈ F p W (u,ψ) W (v,ψ −1 ) f (z) = u(z)u(0)K b (ψ(z)).f (ψ −1 (ψ(z))) = u(z)u(0)K b (az + b)f (z) = |u(0)| 2 K −ab (z)K b (az + b)f (z) = f (z) which also shows that W (u,ψ) W −1 (u,ψ) = I. As shown below and Theorem 3.1, the spectrum of some of these class of operators are contained in the unit circle. The uniformly mean ergodic results in part (ii) of Theorem 4.4 and Theorem 4.5 deal with when ψ(z) = az + b form with |a| = 1 and b = 0. The case for b = 0 is our next point of interest. the spectrum σ(W (u,ψ) ) is contained in the unit circle T. Furthermore as a is not a root of unity , it follows that 1 is an accumulation point of the spectrum of W (u,ψ) .
Moreover 1 is in the spectrum of W (u,ψ) since u(0)e a|b 2 | a−1 a 0 = u b 1−a a 0 = 1. Then an application of Theorem 3.16 of [8] gives the conclusion.
4.1. The multplication operator. We now conclude the section by specializing the main results made in the above section to the multiplication operator M u acting on Fock spaces. Note that from Lemma 2.3 in [21], it is known that the operator M u is bounded on F p if and only if u is a constant function. The same conclusion can be also easily drawn by applying the condition in (2.5) along with Liouville's Theorem.