relation between essential spectrum and spectrum of an operator

Question

Let $T$ be a linear bounded operator from $X \rightarrow X$ where $X$ is Banach space. I want to prove that essential spectrum is subset of spectrum. Where essential spectrum is {$\lambda \in \mathbb{C}$:$\lambda$I-T is not Fredholm operator} and here spectrum has usual meaning.

Answer 1

Notation:

essential spectrum of $T$: $\sigma_e(T)$,

spectrum of $T$: $\sigma(T)$,

$ \rho_e(T)= \mathbb C \setminus \sigma_e(T)$ and $ \rho(T)= \mathbb C \setminus \sigma(T)$.

Let $ \lambda \in \rho(T)$, then $ \lambda I-T$ is invertible, hence $ \lambda I-T$ is Fredholm. Therefore $ \lambda \in \rho_e(T)$.

We have shown: $ \rho(T) \subseteq \rho_e(T)$. It follows:

$$ \sigma_e(T) \subseteq \sigma(T).$$