Weighted/unweighted composition operators which are Ritt or unconditional Ritt operators
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Archiv der Mathematik
Weighted/unweighted composition operators which are Ritt or unconditional Ritt operators Mahesh Kumar
Abstract. In this paper, we study when a composition operator or a weighted composition operator on a Banach space of holomorphic functions is a Ritt operator or an unconditional Ritt operator. It turns out that for composition operators or weighted composition operators on a Banach space of holomorphic functions, if a composition operator or a weighted composition operator is a Ritt operator, then it is also an unconditional Ritt operator. Mathematics Subject Classification. 30D05, 47D03, 47B33. Keywords. Weighted composition operators, Hardy spaces, Weighted Bergman spaces, Weighted Hardy spaces, Ritt operators, Unconditional Ritt operators, Asymptotic behaviour.
1. Introduction. In recent years, Ritt operators have been studied extensively, see, for example, [4,13,19]. Recall that an operator T ∈ L(X), where X is a complex Banach space, is said to be a Ritt operator if rX (T ) ≤ 1 and there exists a constant C > 0 such that R(λ, T ) ≤
C , |λ − 1|
(1)
where rX (T ) (or r(T )) denotes the spectral radius of T ∈ L(X) and R(λ, T ) = (λ−T )−1 is the resolvent operator of T . The condition (1) implies that σX (T )∩ T ⊆ {1}, where σX (T ) is the spectrum of T and T is the unit circle of the complex plane C, and that if T is a Ritt operator, then so is any operator which is similar to T . We recall the following result which will be useful for studying Ritt operators.
M. Kumar
Arch. Math.
Theorem 1.1 (Katznelson–Tzafriri theorem [14, Theorem 1]). Let X be a complex Banach space and let T ∈ L(X) be a power bounded operator. Then lim T n (I − T ) = 0
n→∞
(2)
if and only if σX (T ) ∩ T ⊆ {1}. Here I is the identity operator on X. The following result shows that Ritt operators are precisely those for which the rate of decay in (2) is no slower than n−1 . Theorem 1.2 ([19, Theorem 2.3]). Let X be a complex Banach space. An operator T ∈ L(X) is a Ritt operator if and only if it is power-bounded and T n (I − T ) = O(n−1 ) as n → ∞. Recall that an operator T on a Banach space X is said to satisfy the unconditional Ritt condition, or to be an unconditional Ritt operator (see [13]), if there exists a constant C > 0 such that, for any n ≥ 0 and any a0 , . . . , an ∈ C, n k ak T (I − T ) ≤ C max |ak |. (3) 0≤k≤n k=0
Remark 1.3. (a) The condition (3) is equivalent to having ∞ |L(T n (I − T )x)| ≤ CxL
(4)
n=0
for all x ∈ X, L ∈ X , where X denotes the dual space of X. (b) It follows from [13, Proposition 4.3] that if T satisfies the unconditional Ritt condition (3), then (1) holds, and so T is a Ritt operator. (c) Again, if T satisfies the unconditional Ritt condition, then so does any operator which is similar to T . Let Ω ⊂ C be open and connected, and let Hol(Ω) denote the space of all holomorphic functions on Ω. Recall from [1] that a Banach space X is called a Banach space of holomorphic functions on Ω, and we denote it by X → Hol(Ω), if X ⊂ Hol(Ω) and the
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