The Cesàro operator in weighted ℓ₁ spaces

Abstract

Unlike for ℓp, 1<p≤∞, the discrete Cesàro operator sans-serifC does not map ℓ1 into itself. We identify precisely those weights w such that sans-serifC does map ℓ1false(wfalse) continuously into itself. For these weights a complete description of the eigenvalues and the spectrum of sans-serifC are presented. It is also possible to identify all w such that sans-serifC is a compact operator in ℓ1false(wfalse). The final section investigates the mean ergodic properties of sans-serifC in ℓ1false(wfalse). Many examples are presented in order to supplement the results and to illustrate the phenomena that occur.

Toeplitz operators on the space of all holomorphic functions on finitely connected domains

“…This is explained and studied in Sect. 4. While reading the Introduction the reader should have in mind operators given by (3) for some F ∈ S(D), which can be thought as holomorphic in D close to bD.…”

“…Naturally, in the first case the functions extend holomorphically to ∞ by 0. The symbol spaces are defined as appropriate inductive limits, just as in (4). In order to formulate our main results we need some definitions, which summarize the remarks given so far.…”

“…We only mention the articles which have had some impact on our research. This includes the results by Albanese, Bonet and Ricker [2][3][4][5], Bonet, Lusky and Taskinen [10,11], Bonet and Domański [9], Bonet and Taskinen [12] and also Trybuła [43]. Some of this research concerns Toeplitz operators on weighted sup-norm spaces.…”

Operators on the Fréchet sequence spaces $${\varvec{ces(p+)}}$$ c e s ( p + ) , $$1\le p<\infty $$ 1 ≤ p < ∞

“…The spectra of C 1 have been investigated in various Banach sequence spaces. For instance, we mention ℓ p (1 < p < ∞), [22,23,30,40], c 0 [1,40,51], c [40], ℓ ∞ [50,51], the Bachelis spaces N p (1 < p < ∞) [25], bv and bv 0 [47,48], weighted ℓ p spaces [6,8], the discrete Cesàro spaces ces(p) (for p ∈ {0}∪(1, ∞)), [24], and their dual spaces d s (1 < s < ∞), [19]. For the class of generalized Cesàro operators C t , for t ∈ (0, 1), a study of their spectra and compactness properties (in ℓ 2 ) go back to Rhaly, [52,53].…”