Basic spectral properties of banach algebra

Definition Let {B} be a banach space, {\mathcal{L}(B)} be the space of bounded linear operator from {B \rightarrow B}. If {T \in \mathcal{L}(B)}, then say a complex number {\lambda} is in the resolvent set {\rho(T)} of {T} if {\lambda I - T} has a bounded inverse. Then we define the resolvent of {T} at {\lambda} to be {R_\lambda(T) = (\lambda I -T)^{-1}}. If {\lambda \not\in \rho(T)}, then {\lambda} is said to be in the spectrum {\sigma(T)} of T.

We will learn some properties of the resolvent and the spectrum.

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