The Cesàro operator in weighted ℓ1 spaces. Issue 7 (31st July 2017)

• Record Type: Journal Article
• Authors: Albanese, Angela A.; Bonet, José; Ricker, Werner J.
• Is Part Of: Mathematische Nachrichten. Volume 291:Issue 7(2018)
• Page Start: 1015
• Page End: 1048
• Publication Date: 2017-07-31
• Subjects: Cesàro operator -- compact operator -- mean ergodic operator -- spectrum -- weighted l1 space -- Primary: 47A10 -- 47B37; Secondary: 47A16 -- 47A35 -- 47B07 -- 46B45; Mathematics -- Periodicals; 510.5
• DOI: 10.1002/mana.201600509
• Languages: English
• ISSNs: 0025-584X

Abstract

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