Table 1 Goldberg’s decomposition of the spectrum
(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) | - | \(\lambda \in\sigma_{p} ( T ) \) |
(II) | \(\overline{R(\lambda I-T)}=X\) | λ∈ρ(T) | \(\lambda\in\sigma_{c} ( T ) \) | \(\lambda\in\sigma_{p} ( T ) \) |
(III) | \(\overline{R(\lambda I-T)}\neq X\) | \(\lambda\in\sigma _{r} ( T ) \) | \(\lambda\in\sigma_{r} ( T ) \) | \(\lambda\in \sigma _{p} ( T ) \) |