# 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.

Definition ${X,Y}$ are Banach spaces, we say ${X}$ is embedded in ${Y}$, denoted by ${X \hookrightarrow Y}$, if ${X \subset Y}$ and ${\| \cdot \|_Y \le C \| \cdot \|_X}$.
Notation: ${\langle \xi \rangle := \sqrt{1+\xi^2}}$.