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Table 2 Separations of the spectrum [ 24 ]

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) - \(\lambda\in\sigma_{p} ( T ) \)
\(\lambda\in\sigma_{ap} ( T ) \)
(II) R(λI − T)≠X
\(\overline {R ( \lambda I-T ) }=X\)
λρ(T) \(\lambda\in\sigma_{c} (T ) \)
\(\lambda\in\sigma _{ap} ( T ) \)
\(\lambda\in\sigma_{\delta} ( T ) \)
\(\lambda\in\sigma_{p} ( T ) \)
\(\lambda \in\sigma_{ap} ( T ) \)
\(\lambda\in\sigma_{\delta } ( T ) \)
(III) \(\overline{R ( \lambda I-T ) }\neq X\) \(\lambda\in\sigma_{r} ( T ) \)
\(\lambda\in\sigma _{\delta} ( T ) \)
\(\lambda\in\sigma_{co} ( T ) \)
\(\lambda\in\sigma_{r} ( T ) \)
\(\lambda \in\sigma_{ap} ( T ) \)
\(\lambda\in\sigma_{\delta } ( T ) \)
\(\lambda\in\sigma_{co} ( T ) \)
\(\lambda\in\sigma_{p} ( T ) \)
\(\lambda\in\sigma _{ap} ( T ) \)
\(\lambda\in\sigma_{\delta} ( T ) \)
\(\lambda\in\sigma_{co} ( T ) \)