Power convergence of Abel averages

Necessary and sufficient conditions are presented for the Abel averages of discrete and strongly continuous semigroups, $T^k$ and $T_t$, to be power convergent in the operator norm in a complex Banach space. These results cover also the case where $T$ is unbounded and the corresponding Abel average is defined by means of the resolvent of $T$. They complement the classical results by Michael Lin establishing sufficient conditions for the corresponding convergence for a bounded $T$.


Posing the problem
For a bounded linear operator T on a Banach space X, the Abel average of the discrete semigroup {T k } k∈N0 is defined as where α is a suitable numerical parameter, i.e., such that A α belongs to L(X)the Banach algebra of all bounded linear operators 1 on X.
Likewise, for a strongly continuous semigroup {T t } t≥0 , the Abel average is defined by the formula (1.2)Ã λ = λ ∞ 0 e −λs T s ds, with a suitable parameter λ, which is to be understood point-wise, as an improper Riemann integral; see, e.g., [5, page 42].
In this note, we establish necessary and sufficient conditions which ensure that the averages (1.1) and (1.2) are power convergent in the operator norm. Our main result (Theorem 2.1 below) covers also the case where T in (1.1) is unbounded.
The study of the Abel averages goes back to at least E. Hille [7] and W.F. Eberlein [4]. They are presented in the books [5,8,11,18]. Uniform ergodic theorems for Abel and Cesàro averages were established by M. Lin in [13] and [14]. The following assertions can be deduced from the corresponding nowadays classical results of [14]. Assertion 1.1. Let T be such that (1.3) T n /n → 0, as n → ∞.
Then, for each α ∈ (0, 1), the operator A α in (1.1) belongs to L(X), and the following statements are equivalent: and let B be its generator. Then, for all λ > 0, the operatorÃ λ in (1.2) is in L(X) and the following statements are equivalent: The limits in (ii) and (iii) coincide; their common value is the projectionẼ of X onto KerB along ImB, given by the Riesz decomposition 2 corresponding to the (at most) simple pole 0 of the resolvent of B.

Note that
where the inclusion " ⊂ " follows by, e.g., [18,Theorem 1.8.3,page 33]. In the discrete case, an analog of claim (iii) of Assertion 1.2 can also be obtained. As follows from (1.3), the spectrum of T is contained in the closure of the open unit disk ∆. By the spectral mapping theorem, the spectrum of A α is then contained in ∆ ∪ {1}. Since and cf. the proof of Theorem 2.1 below, we have the Riesz decomposition and thus the point 1 is at most a simple pole of A α . In particular, it is at most an isolated point of the spectrum of A α . Hence, A n α /n → 0, as n → +∞; see, e.g., [16]. Therefore, all the operators A α , α ∈ (0, 1), are uniformly ergodic, even power convergent to the same limit E as above. This complements Assertion 1.1 in the spirit of Assertion 1.2. As we shall see in Assertions 1.3 and 1.4 below, both claims (ii) above are equivalent to the power convergence of the corresponding Abel averages; see also Remark 2.2 below. Indeed, under the conditions of Assertions 1.1 and 1.2, by the technique used in [14] one can show that, for α close to 1 − and λ close to 0 + , the operators A α andÃ λ , respectively, are power convergent in L(X). As we shall see later, if X is a complex Banach space, the assumptions of Assertions 1.1 and 1.2 allow one to prove the corresponding power convergence of the operators A α and A λ , for all α ∈ (0, 1) and all λ > 0, respectively. More precisely, the following extensions of Assertions 1.1 and 1.2 hold. See also [15]. Assertion 1.3. Let T be a bounded linear operator in a complex Banach space X obeying (1.3), and let A α , α ∈ (0, 1), be its Abel average (1.1). Then the following statements are equivalent:

strongly continuous semigroup of bounded linear operators in a complex
Banach space X such that (1.4) holds. Let B be its generator andÃ λ , λ > 0, be its Abel average (1.2). Then the following statements are equivalent: B has closed range; (ii) for some λ > 0, the sequence {Ã n λ } n∈N converges in L(X); (iii) for each λ > 0, the sequence {Ã n λ } n∈N converges in L(X).

The limits in (ii) and (iii) coincide with the projectionẼ from Assertion 1.2.
In fact, the conditions (1.3) and (1.4) are quite far from being necessary for the corresponding Abel averages to converge as stated above. For example, the former one can be replaced by the dissipativity condition used in the classical Lumer-Phillips theorem; see, e.g., [23, page 250]. The next assertion, which provides an example of this sort, might be useful in the study of the sets of fixed points of some nonlinear operators; see [19] and [20].
which, by [17,Theorem 3.1], is equivalent to Indeed, by [6], condition (1.6) and the closedness of (I − T )X yield the existence of lim α→1 − A α , which is equivalent to the fact that the point 1 is at most a simple pole of the resolvent of T ; see [

The results
In this section, we derive the conditions that are necessary and sufficient for the statements of Assertions 1.3, 1.4, and 1.5 to hold. Moreover, our results cover also the case where T in (1.1) is unbounded, and hence (1.3) is not applicable. The key observation which allowed us to get them is that the principal thing one needs is the spectrum σ(T ) lying merely in the half-plane Π = {ζ ∈ C : Reζ ≤ 1}.
Note also that (1.3) and statement (i) in Assertion 1.3 imply that see, e.g., [16] and [19, pages 40-43]. In the sequel, for a closed densely defined linear operator T in a complex Banach space X, by D(T ) and ρ(T ) we denote the domain and the resolvent set of T , respectively. For such an operator with (1, +∞) ⊂ ρ(T ), the Abel average can be defined as the following bounded linear operator Finally, by Im(I − T ) we mean the (I − T )-image of D(T ).
For every α ∈ (0, 1), the limit in (i) is the projection of X onto Ker(I − T ) along Im(I − T ).

It maps the domain Ω
Remark 2.2. Condition (2.1) in (ii) of Theorem 2.1 can be replaced by the existence of lim α→1 − A α . In view of (2.4) and (2.5), the latter limit is equal to the Riesz projection E of X onto Ker(I − T ) along Im(I − T ), given by the decomposition in (2.1). The point 1 is simultaneously at most a simple pole of the resolvents of both T and A α .
The theorem just proven obviously extends Assertion 1.3. Since the closure of the numerical range of a bounded linear operator contains its spectrum (see, e.g., (iii) for each λ > 0, the sequence {Ã n λ } n∈N converges in L(X). For each λ > 0, the limit above is the projection of X onto KerB along ImB.
With the help of [3, Theorem VIII.1.11, page 622] we get the following generalization of Assertion 1.4. Recall that the Abel averageÃ λ was defined in (1.2) and its n-th power can be written as (see, e.g., [5, page 43 As mentioned just after Assertion 1.2, we have Corollary 2.4. Let {T t } t≥0 be a strongly continuous semigroup in a complex Banach space X. Let B be its generator andÃ λ be its Abel average (1.2). Assume also that Then the following statements are equivalent: (ii) for some λ > 0, the sequence {Ã n λ } n∈N converges in L(X); (iii) for each λ > 0, the sequence {Ã n λ } n∈N converges in L(X). For each λ > 0, the limit in (ii) and (iii) is the projection of X onto KerB along ImB.

An example
We present an unbounded linear operator T , which has the properties described by Theorem 2.1. Here X is the complex Hilbert space L 2 (R).
where S(R) is the space of Schwartz test functions. Then T 0 is essentially selfadjoint and such that (3.2) T 0 x n = λ n x n , λ n = 1 − 2n, n ∈ N 0 .
The eigenvalues λ n are simple and the eigenvectors constitute an orthonormal basis of X; see, e.g., [1, pages 36-39]. In (3.3), for n ∈ N 0 , h n is the Hermite polynomial of degree n. In particular, h 0 = π 1/4 . Let T be the closure of (3.1). Then X 0 := Ker(I − T ) is the one-dimensional subspace of X spanned by x 0 . Let X 1 be the orthogonal complement of X 0 , i.e., (3.4) X = X 0 ⊕ X 1 .
Take any x ∈ X 1 . Then Thus, in view of (3.2), R(λ, T ) is a compact operator, positive for λ > 1. Then its spectral decomposition is where P n , n ∈ N 0 , is the orthogonal projection onto the subspace spanned by x n . For λ > 1 and any x ∈ X, cf. (3.5), we have which yields that, in L(X), (λ − 1)R(λ, T ) → P 0 as λ → 1 + . For m ∈ N, by (3.6) we have Then, for m ≥ 4,