Bishop's Property (β) and the Cesàro Operator

Spectral properties of generalized Cesàro operators on Hardy and weighted Bergman spaces

“…Moreover, there is an intrinsic characterization of subdecomposability, [1], which leads to a sufficient condition in terms of growth of left resolvent functions that is often applicable to operators with thin approximate point spectrum, [15,Section 1.7]. As shown in [20] and [6], such a condition applies to the Cesàro operator C 1 | L p,α a for all p > 1 and all α −1. Therefore, in contrast to hyponormality, C 1 | L p,α a is always subdecomposable.…”

“…If |λ| > 0, let D(λ) = V (λ, |λ|). The following theorem is due to Siskakis, Miller, Miller and Smith in the setting of Hardy spaces and unweighted Bergman spaces and due to Dahlner in the weighted Bergman space case, [23], [26], [20], [17], [18] and [6]. The case p = 1 remains open for all α −1.…”

The Cesàro and related operators, a survey

“…Siskakis's techniques were later exploited in [30] and [27] to obtain a complete spectral picture of the Cesàro operator on the Hardy spaces H p (D), for p > 1, and on the Bergman spaces L p,0 a for p > 2. However, due to the existence of eigenvalues, it seems to be impossible to go beyond the limiting case p = 2 in the Bergman space setting using the methods of composition semigroups.…”

“…The reason for this is that one needs resolvent estimates around the eigenvalues, while the composition semigroups only give such estimates for larger regions of the spectrum. Dahlner, [10], uses an alternative approach to replicate the spectral results of [30] and [27] for C| L p,α a , for all p and α, 1 < p, −1 < α. Very recently, Anna-Maria Persson has shown in her PhD thesis under Alemann's direction that the Cesàro operator is subdecomposable on H 1 and on the standard weighted Bergman spaces L 1,α a , α ≥ 0.…”

“…If |λ| > 0, let D(λ) = V (λ, |λ|). The following theorem summarizes the spectral properties of the Cesàro operator on the spaces L p,α a , [35], [37], [30], [27], [28] and [10].…”