THE CESÀRO OPERATOR IN THE FRÉCHET SPACES ℓ^p+ANDL^p−

Abstract

The classical spaces ℓp+, 1 ≤ p < ∞, and Lp−, 1<p ≤ ∞, are countably normed, reflexive Fréchet spaces in which the Cesàro operator C acts continuously. A detailed investigation is made of various operator theoretic properties of C (e.g., spectrum, point spectrum, mean ergodicity) as well as certain aspects concerning the dynamics of C (e.g., hypercyclic, supercyclic, chaos). This complements the results of [3, 4], where C was studied in the spaces ℂℕ, Lploc(ℝ+) for 1 < p < ∞ and C(ℝ+), which belong to a very different collection of Fréchet spaces, called quojections; these are automatically Banach spaces whenever they admit a continuous norm.

The Cesàro operator on smooth sequence spaces of finite type

“…. , 1 s }, and condition (3) in Theorem 1 does not imply condition (4). This is easy to show since (i s−1 ) i ∈ λ 0 (A), hence (i m−1 ) i ∈ λ 0 (A) for m = 1, 2, .…”

“…We refer the reader to the introduction of [3]. The behaviour of C when acting on the Fréchet spaces C N , ℓ p+ = q>p ℓ q , 1 ≤ p < ∞, and on the power series space Λ 0 (α) of finite type was studied in [2,4,5]. In this paper we extend the results of [5] to the strictly more general setting of the smooth sequence spaces λ 0 (A) of finite type.…”

The Cesàro operator in weighted ℓ1 spaces

“…We begin by recalling the following known fact; see e.g. [, Proposition 4.1], [, Propositions 4.3 and 4.4]. For convenience of notation we set and .…”

“…Proof It is known that is power bounded, uniformly mean ergodic and satisfies both and see [, Proposition 4.1], [, Proposition 4.3]. Observe that the sequence converges to in .…”

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

“…Then C ∈ L(k 0 (V )) is not supercyclic, and hence not hypercyclic either.Proof. Follows from[4, Proposition 4.3] and[5, Proposition 4.4].…”