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Table 1 Goldberg’s decomposition of the spectrum

From: The spectrum and some subdivisions of the spectrum of discrete generalized Cesàro operators on \(\ell_{p}\) (\(1< p<\infty\))

   (1) (2) (3)
R ( λ ; T ) exists and is bounded R ( λ ; T ) exists and is unbounded R ( λ ; T ) does not exists
(I) R(λI − T)=X λρ(T) - \(\lambda \in\sigma_{p} ( T ) \)
(II) \(\overline{R(\lambda I-T)}=X\) λρ(T) \(\lambda\in\sigma_{c} ( T ) \) \(\lambda\in\sigma_{p} ( T ) \)
(III) \(\overline{R(\lambda I-T)}\neq X\) \(\lambda\in\sigma _{r} ( T ) \) \(\lambda\in\sigma_{r} ( T ) \) \(\lambda\in \sigma _{p} ( T ) \)