Definition of essential spectrum?

Question

Suppose we have a Hilbert space $\mathscr{H}$ and a bounded linear map $T\in\mathscr{B(H)}$ NOT necessarily self-adjoint. There seems to be loads of definitions of the essential spectrum of $T$. My question is, whether in the Hilbert space setting the following are equivalent:

1. $\lambda$ is such that $T-\lambda{I}$ is not Fredholm

2. $\lambda$ is in $\sigma(T)\backslash\sigma_{d}(T)$ where $\sigma_{d}(T)$ denotes the set of isolated points of the spectrum (discrete spectrum).

Answer 1

1. Semi Fredholm class is $\mathcal{SF}$, which is the union of left semi Fredholm class and right semi Fredholm class. Fredholm class is $\mathcal{F}$, which is the intersection between left semi Fredholm class and right semi Fredholm class.
2. Fredholm index: $\mbox{ind}(T)$ of $T\in\mathcal{SF}$ (Semi Fredholm) is defined by $\mbox{ind}(T)=\dim \ker(T)-\dim \ker(T^*).$
3. $\sigma_0(T)=\{\lambda \in \sigma(T): \lambda I-T\in \mathcal{F} \mbox{ and ind}(\lambda I-T)=0\}$.
4. Let $T\in \mathcal{B}(\mathcal{H})$ and $\sigma_w(T)$ be the Weyl spectrum of $T$. Then $\sigma(T)=\sigma_w(T)\cup \sigma_0(T)$.
5. Let $T$ to be self adjoint operator. Then $\sigma_d(T)=\sigma_0(T)$ and $\sigma_w(T)=\sigma_e(T)$. Therefore $\sigma(T)=\sigma_e(T)\cup \sigma_d(T)$. Here $\sigma_e(T)$ is the essential spectrum of $T$.