I have been introduced to the following definition of the essential numerical range of a bounded, linear operator on a separable, infinite-dimensional Hilbert space: $$W_e(T) := \bigcap_{K\in \mathcal K(H)} \overline{W(T+K)}$$ i.e., the essential numerical range $W_e(T)$ of $T$ is the intersection of the closures of numerical ranges of all compact perturbations of $T$.
- Why is the essential numerical range defined in this way, i.e., what motivates this definition and how is it useful? Even in my wildest dreams, I would have no reason to consider compact perturbations of $T$ in order to define $W_e(T)$.
I have noticed that definitions in operator theory on infinite-dimensional Hilbert spaces are often motivated by finite-dimensional analogs, but I'm not able to draw any connections here. Thanks for any insights!
Note: The numerical range of a (bounded, linear) operator $T\in \mathcal B(\mathcal H)$ is defined as $$W(T) := \{\langle Tx,x\rangle: x\in \mathcal H, \|x\| = 1 \}$$
First of all, I am not an expert on this topic, but let me try and provide you with some insight.
One of the application of the numerical range is a rough approximation of spectrum. In particular, for $T\in \mathcal{B}({\mathcal{H})}$ closure of the numerical range is smallest convex set (i.e. convex hull) that contains the spectrum of $T$
$$\operatorname{conv}(\sigma(T))= \overline{W(T)}. \quad (*)$$
Now, for a self-adjoint operator there is a notion of essential spectrum:
$$\sigma_e(T)=\{\lambda\in\mathbb{C}: T-\lambda I \text{ is not Fredholm }\}.$$
For $\mathcal{H}$ finite-dimensional the spectrum is always discrete and essential spectrum is empty, so in a way essential spectrum captures the infinite-dimensional part of the spectrum or the part of the spectrum which is not discrete (i.e. not finite dimensional).
There are several equivalent characterizations of essential spectrum for a self-adjoint case, each of which provides inequivalent definitions of essential spectrum for general bounded linear operator $T\in\mathcal{B}(\mathcal{H})$. Let's focus on two (here I use Wikipedia's notation):
$$\sigma_{e,4}(T)=\bigcap_{K\in\mathcal{K}(H)}\sigma(A+K),\\ \sigma_{e,3}(T)=\{\lambda\in\mathbb{C}: T-\lambda I \text{ is not Fredholm}\}.$$
For self-adjoint operator two sets coincide and for general $T\in\mathcal{B}(\mathcal{H})$ $$\sigma_{e,4}=\sigma_{e,3}\cup \{\lambda\in\mathbb{C}: T-\lambda I \text{ is Fredholm and } \operatorname{ind}(T-\lambda I)\neq 0\}. \quad (**)$$
It should be noted that $\sigma_{e,4}(T)$ is the largest subset of spectrum $T$ invariant under any compact perturbations and $\sigma_{e,3}$ is the complement of the set for which $ T-\lambda I$ is invertible modulo compact operator (Atkinson's theorem). Both of the properties hold for the spectrum of self-adjoint operator and are quite natural for the intuitive definition of essential spectrum as "non-finite-dimensional part".
Lastly, using $(**)$ and continuity of index it can be shown that $$\operatorname{conv}(\sigma_{e,4})=\operatorname{conv}(\sigma_{e,3}).$$
So what would be the appropriate definition of essential spectrum that would approximate essential (i.e. non-finite) part of the spectrum in the sense of $(*)$?
Atkinson's theorem gives us another characterization of $\sigma_{e,3}(T)$: $$\sigma_{e,3}(T) = \sigma([T]),$$ where $[T]$ is equivalence class of $T$ in the Calkin algebra $\mathcal{B}(\mathcal{H})/\mathcal{K}(\mathcal{H})$. There's a notion of a numerical range for a unital Banach algebra $\mathcal{A}$, which generalizes the usual numerical range: $$W_0(a)=\{f(a): f \text{ is a state of }\mathcal{A} \}.$$ Thus we can define a numerical range for an element of Calkin algebra $[T]$, which was originally called essential numerical range: $$W_e(T):=W_0([T]).$$ Note that since Calkin algebra is a $C^*$-algebra, it can be thought of as a $C^*$-subalgebra of $\mathcal{B}(\tilde{\mathcal{H}})$ for some Hilbert space $\mathcal{H}$ with the usual definition of numerical range, which coincides with $W_0$. Thus $$\overline{W_e(T)}=\operatorname{conv}(\sigma([T]) = \operatorname{conv}(\sigma_{e,3}(T))=\operatorname{conv}(\sigma_{e,4}(T)).$$
Moreover, it can be shown that $$W_e(T)=\bigcap_{K\in\mathcal{K}(\mathcal{H})}\overline{W(T+K)}$$ and this formula can be used as a definition which is quite natural in that sense and for essential numerical range of bounded linear operator there's only one natural definition, unlike the case of essential spectrum.
Summing up, essential numerical range is specifically about infiniteness and finite-dimensional analogs can't be found. Essential numerical range was introduced in GROWTH CONDITIONS AND THE NUMERICAL RANGE IN A BANACH ALGEBRA by J. G. STAMPFLI and J. P. WILLIAMS. This answer is based on a paper On the essential numerical range, the essential spectrum, and a problem of Halmos by P. A. FILLMORE, J. G. STAMPFLI, and J. P. WILLIAMS which is a good survey on the subject. It contains the proofs for all of the statements above as well as other equivalent characterizations of essential numerical range which can be used to generalize it to the case of unbounded linear operators:
$$W_e(T)=\{\lambda \in \mathbb{C}: \exists\ \{x_n\} \text{ s.t. } \|x_n\|=1, x_n\overset{w}{\to} 0, (Tx_n,x_n)\to\lambda\}\\= \{\lambda\in \mathbb{C}: (Te_n,e_n)\to\lambda \text{ for some orhtonormal sequence}\} \\ =\{\lambda \in \mathbb{C}: PTP-\lambda P \in \mathcal{K}(\mathcal{H}) \text{ for some infinite-rank projection } P\}.$$