Table 2 Separations of the spectrum [ 24 ]
(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 | λ∈ρ(T) | - | \(\lambda\in\sigma_{p} ( T ) \) \(\lambda\in\sigma_{ap} ( T ) \) |
(II) | R(λI − T)≠X \(\overline {R ( \lambda I-T ) }=X\) | λ∈ρ(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) | \(\overline{R ( \lambda I-T ) }\neq X\) | \(\lambda\in\sigma_{r} ( T ) \) \(\lambda\in\sigma _{\delta} ( T ) \) \(\lambda\in\sigma_{co} ( T ) \) | \(\lambda\in\sigma_{r} ( T ) \) \(\lambda \in\sigma_{ap} ( T ) \) \(\lambda\in\sigma_{\delta } ( T ) \) \(\lambda\in\sigma_{co} ( T ) \) | \(\lambda\in\sigma_{p} ( T ) \) \(\lambda\in\sigma _{ap} ( T ) \) \(\lambda\in\sigma_{\delta} ( T ) \) \(\lambda\in\sigma_{co} ( T ) \) |