The Cesàro operator on smooth sequence spaces of finite type

Abstract

The discrete Cesàro operator C is investigated in the class of smooth sequence spaces λ 0 (A) of finite type. This class contains properly the power series spaces of finite type. Of main interest is its spectrum, which is distinctly different in the cases when λ 0 (A) is nuclear and when it is not. The nuclearity of λ 0 (A) is characterized via certain properties of the spectrum of C. Moreover, C is always power bounded and uniformly mean ergodic on λ 0 (A).

The Cesàro Operator on Duals of Smooth Sequence Spaces of Infinite Type

“…Proof. The proof reads as [15,Proposition 9]. In the implication (1) ⇒ (2) if we set b n (i) := n j=1 a j (i), the rest follows with the same arguments.…”

“…The Köthe echelon space λ 1 (A) of order 1 is said to be a smooth sequence space of finite type (or a G 1 -space) [21, Section 3] if A satisfies (G1-1) 0 < a n (i + 1) ≤ a n (i), for all n ∈ N and i ∈ N. (G1-2) For all n ∈ N there exist m > n and C > 0 such that a n (i) ≤ Ca m (i) 2 , for all i ∈ N. The Cesàro operator on G 1 -spaces was studied by the author in [15]. The following proposition shows that C is not continuous on duals of nuclear G 1 -spaces.…”

“…For a proof, see e.g. [15,Lemma 16]. := 0 for i > s. It is straightforward to show that u (s) ∈ k 0 (V ) ′ (since u (s) belongs to the space of finitely supported sequences c 00 ) and C ′ u (s) = 1 s u (s) , that is, λ ∈ σ pt (C ′ , k 0 (V ) ′ ).…”

On the spectrum of isomorphisms defined on the space of smooth functions which are flat at 0

“…The study of the spectrum of operators defined on Fréchet spaces or more general locally convex spaces has been an object of research in the last years, see e.g. [2,3,5,6,10,12,13,16,18].…”